Chapter 2 — Lines, Linear Functions and Systems

Answer: B

Explanation

Label the points and subtract in a consistent order. Take $(2, 3)$ as the first and $(6, 11)$ as the second:

$$m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$$

Check it makes sense: as $x$ goes from 2 to 6 the point rises from 3 to 11, so the line goes up — and the slope is positive. ✓

It does not matter which point you call first, as long as you are consistent. Starting from $(6, 11)$ instead:

$$m = \frac{3 - 11}{2 - 6} = \frac{-8}{-4} = 2$$

Same answer. The two minus signs cancel, which is exactly why consistency is the only rule.

Why each wrong option is wrong:

• A, $\frac{1}{2}$ — the change in $x$ over the change in $y$, upside down. Slope is rise over run: $y$ on top.
• B is the answer.
• C, $-2$ — subtracted the $y$ values in one order and the $x$ values in the other. That is the inconsistency warned about above, and it always produces the right number with the wrong sign.
• D, $\frac{7}{4}$ — added the coordinates instead of subtracting them. Slope is about change, so it is always a subtraction.

Takeaway: $m = \frac{y_2 - y_1}{x_2 - x_1}$, with $y$ on top and the same order on both. Then check the sign against whether the line should be rising or falling.

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Answer: D

Explanation

"How much per year" is a slope. The two points are $(2, 140)$ and $(8, 260)$, where $x$ is years and $y$ is height:

$$m = \frac{260 - 140}{8 - 2} = \frac{120}{6} = 20$$

The tree grows 20 cm a year.

Check it: from year 2, six years pass to reach year 8, and $6 \times 20 = 120$ cm of growth — which takes 140 cm up to 260 cm ✓

Why each wrong option is wrong:

• B, $12$ — added the two years instead of subtracting them. The gap is 6 years, not 10.
• C, $\frac{65}{2}$ — divided a height by a year, which is what you would do if the tree had been 0 cm tall at year 0. It was not; it was 140 cm at year 2.
• A, $60$ — divided the growth by 2 rather than by the 6-year gap.

Takeaway: "Per year", "per hour", "for each" all mean slope. Use the gap between the two $x$ values, never one of the values on its own.

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Answer: D

Explanation

Both points have the same $y$-coordinate, 5. So there is no rise at all:

$$m = \frac{5 - 5}{4 - (-3)} = \frac{0}{7} = 0$$

The line is horizontal — it is the line $y = 5$. Walk along it and you never go up or down, so the slope is zero.

Be careful with the pair this question sits next to. Zero on top gives slope 0; zero on the bottom gives an undefined slope, and that is a vertical line like $x = 5$. They sound similar and they are opposites.

$y$ values equal → $\frac{0}{\text{something}}$	horizontal, slope $0$
$x$ values equal → $\frac{\text{something}}{0}$	vertical, slope undefined

Why each wrong option is wrong:

• A, $1$ — there is no default slope; it has to be computed.
• B, $7$ — the horizontal distance between the points, which is the denominator, not the slope.
• C, $\frac{5}{7}$ — used the $y$-coordinate 5 as though it were the rise. The rise is the change in $y$, which here is zero.

Takeaway: Same $y$ means horizontal and slope 0. Same $x$ means vertical and slope undefined. Check whether the coordinates repeat before reaching for the formula.

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Section 2 — The Forms of a Line

The same line can be written three ways. The test moves between them constantly, and each form is convenient for a different question.

Form	Looks like	Read off it immediately
Slope-intercept	$y = mx + b$	slope $m$, $y$-intercept $b$
Point-slope	$y - y_1 = m(x - x_1)$	slope $m$, a point $(x_1, y_1)$
Standard	$Ax + By = C$	both intercepts, quickly

Slope-intercept is the one to convert into. If a question gives you $3x + 4y = 12$ and asks for the slope, rearrange to $y = -\frac34 x + 3$ and read it off. Guessing the slope from the standard form's coefficients is where marks disappear — the slope of $Ax + By = C$ is $-\frac{A}{B}$, not $\frac{A}{B}$ and not $A$.

Finding intercepts is worth its own line, because it is quick and the test asks often:

• $y$-intercept: set $x = 0$.
• $x$-intercept: set $y = 0$.

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Answer: D

Explanation

Rearrange into $y = mx + b$. Subtract $3x$ from both sides:

$$4y = -3x + 12$$

Divide everything by 4:

$$y = -\frac{3}{4}x + 3$$

The slope is the number multiplying $x$:

$$m = -\frac{3}{4}$$

Check the sign by thinking about the picture. When $x$ grows, $3x$ grows, so $4y$ must shrink to keep the total at 12. The line falls, so the slope is negative ✓

Why each wrong option is wrong:

• B, $\frac{3}{4}$ — the right fraction with the minus dropped. The minus appears when $3x$ moves across the equals sign, and it is the single most common slip in this topic.
• C, $-3$ — took the coefficient of $x$ and never divided by 4.
• A, $- \frac{4}{3}$ — inverted the fraction. That is the slope of a perpendicular line, which is the next section, and it is offered here on purpose.

Takeaway: Rearrange into $y = mx + b$ rather than guessing from $Ax + By = C$. If you want the shortcut, the slope is $-\frac{A}{B}$ — but rearranging is safer and takes one extra line.

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Answer: B

Explanation

Find the slope first:

$$m = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3$$

So the line is $y = 3x + b$. Put one of the points in to find $b$ — use $(1, 4)$:

$$4 = 3(1) + b \quad\Longrightarrow\quad b = 1$$

The line is $y = 3x + 1$, and the $y$-intercept is $1$.

Check with the other point, which is the whole value of having two: $3(3) + 1 = 10$ ✓

Why each wrong option is wrong:

• A, $4$ — read a $y$-coordinate straight off one of the given points. The $y$-intercept is where the line crosses at $x = 0$, and neither given point is there.
• C, $3$ — the slope, not the intercept. Both numbers appear in $y = 3x + 1$; the question asked for the second one.
• D, $-2$ — substituted the point's coordinates into the wrong slots.

Takeaway: Slope first, then substitute either point to find $b$ — and check with the other one. Two points give you a free check; use it.

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Answer: 10

Explanation

Crossing the $x$-axis means $y = 0$ there. That is the whole question — once you know it, the algebra is one line.

Set $y = 0$:

$$2x - 5(0) = 20 \quad\Longrightarrow\quad 2x = 20 \quad\Longrightarrow\quad x = 10$$

So the line crosses at $(10, 0)$, and $k = 10$.

Check: $2(10) - 5(0) = 20$ ✓

The pair worth memorising, because it is easy to swap them under pressure:

• On the $x$-axis, $y = 0$.
• On the $y$-axis, $x = 0$.

Each axis is where the other coordinate vanishes. If you had set $x = 0$ here you would have got $-5y = 20$, so $y = -4$ — a real point on the line, and the wrong one.

This is a grid-in. Enter 10.

Takeaway: To find where a line meets an axis, set the other variable to zero. Say which one out loud before you write, because swapping them gives a plausible wrong answer rather than an obviously wrong one.

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Section 3 — Parallel and Perpendicular

Two rules, and the second is the one that gets forgotten.

Lines	Their slopes
Parallel	equal: $m_2 = m_1$
Perpendicular	negative reciprocals: $m_2 = -\dfrac{1}{m_1}$

"Negative reciprocal" means do two things: flip the fraction over, and change the sign. If one slope is $\frac{2}{3}$, the perpendicular slope is $-\frac{3}{2}$. Doing only one of the two is what the wrong options are made of.

A useful check: two perpendicular slopes always multiply to $-1$. $\frac23 \times -\frac32 = -1$ ✓

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Answer: D

Explanation

The slope of $\ell$ is $\frac{2}{5}$. A perpendicular slope is the negative reciprocal — flip it and change its sign:

$$\text{flip: } \frac{5}{2} \qquad \text{change sign: } -\frac{5}{2}$$

Check with the multiplication test:

$$\frac{2}{5} \times \left(-\frac{5}{2}\right) = -\frac{10}{10} = -1 \; ✓$$

Why each wrong option is wrong:

• A, $\frac{2}{5}$ — the same slope, which gives a parallel line, not a perpendicular one.
• B, $\frac{5}{2}$ — flipped the fraction and forgot the sign. $\frac25 \times \frac52 = +1$, not $-1$.
• C, $- \frac{2}{5}$ — changed the sign and forgot to flip. $\frac25 \times -\frac25 = -\frac{4}{25}$, not $-1$.

Options B and C are the two halves of the job, offered separately. Doing both is the entire question.

Takeaway: Perpendicular means flip and negate. Multiply the two slopes together as a check — if you do not get exactly $-1$, you have done one half of the job.

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Answer: B

Explanation

Parallel lines have the same slope, so start by finding the slope of $p$. Rearrange $4x + 3y = 9$:

$$3y = -4x + 9 \quad\Longrightarrow\quad y = -\frac{4}{3}x + 3$$

So $m = -\frac{4}{3}$, and line $q$ is

$$y = -\frac{4}{3}x + b$$

Now use the point $(6, -1)$ that $q$ passes through:

$$-1 = -\frac{4}{3}(6) + b \quad\Longrightarrow\quad -1 = -8 + b \quad\Longrightarrow\quad b = 7$$

Check: at $x = 6$, $-\frac43(6) + 7 = -8 + 7 = -1$ ✓

Why each wrong option is wrong:

• A, $-9$ — used $+\frac43$ as the slope, losing the minus sign when rearranging.
• C, $3$ — the $y$-intercept of line $p$. Parallel lines share a slope, not an intercept; if they shared both they would be the same line.
• D, $- \frac{4}{3}$ — the slope, where the question asked for the intercept.

Takeaway: Parallel means copy the slope, then use the given point to find the new intercept. The old intercept is never the answer — that is what makes the two lines different lines.

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Section 4 — Systems of Two Equations

A system is two equations that must both be true at once. Its solution is the point $(x, y)$ where the two lines cross.

There are two methods, and it is worth being fluent in both because questions are built to favour one or the other.

Substitution — best when one equation already has a variable alone. Rearrange one equation to "$y = \ldots$", put that into the other, solve, then substitute back.

Elimination — best when the equations are lined up as $Ax + By = C$. Add or subtract the two equations so that one variable cancels. Multiply one or both equations first if nothing cancels straight away.

Always find both values, and always check both equations. Half the wrong options on system questions are the other variable's value.

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Answer: B

Explanation

The first equation already gives $y$ on its own, so substitute it into the second:

$$3x + (2x - 1) = 14$$

Collect:

$$5x - 1 = 14 \quad\Longrightarrow\quad 5x = 15 \quad\Longrightarrow\quad x = 3$$

Then $y = 2(3) - 1 = 5$, so the crossing point is $(3, 5)$.

Check in both equations: $y = 2(3) - 1 = 5$ ✓ and $3(3) + 5 = 14$ ✓

Why each wrong option is wrong:

• A, $2$ — dropped the $-1$ when substituting, solving $5x = 10$.
• C, $5$ — the value of $y$. Both 3 and 5 come out of this question, and the last line asked for $x$.
• D, $\frac{13}{5}$ — moved the $-1$ to the wrong side, solving $5x = 13$.

Takeaway: When one equation is already "$y = \ldots$", substitute. Then check in both equations — an answer that satisfies only one of them is not a solution to the system.

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Answer: D

Explanation

Both equations have $2x$, so subtracting them removes $x$ in one step:

$$(2x + 3y) - (2x - y) = 16 - 4$$

On the left, $2x - 2x = 0$ and $3y - (-y) = 4y$:

$$4y = 12 \quad\Longrightarrow\quad y = 3$$

Then from the second equation, $2x - 3 = 4$, so $2x = 7$ and $x = \frac72$.

Check both: $2(\frac72) + 3(3) = 7 + 9 = 16$ ✓ and $2(\frac72) - 3 = 4$ ✓

The step people get wrong is $3y - (-y)$. Subtracting a negative adds, so it is $4y$, not $2y$. Writing the subtraction out with brackets, as above, prevents it.

Why each wrong option is wrong:

• B, $\frac{7}{2}$ — the value of $x$. Again, the question named one of the two.
• C, $5$ — added the equations instead of subtracting. That gives $4x + 2y = 20$, which eliminates nothing.
• A, $20$ — a stray total rather than a value of $y$; if $4y = 12$ then $y$ is 3, and 20 never enters a correct solution.

Takeaway: Line the equations up and look for a variable with matching coefficients. Same sign, subtract; opposite signs, add. And bracket the second equation before subtracting it, so the double negative does not bite.

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Answer: C

Explanation

Let $c$ be the price of a coffee and $t$ the price of a tea. The two sentences become two equations:

$$3c + 2t = 16 \qquad c + 4t = 12$$

Eliminate $t$. Multiply the first equation by 2 so both have $4t$:

$$6c + 4t = 32$$

Now subtract the second equation:

$$(6c + 4t) - (c + 4t) = 32 - 12 \quad\Longrightarrow\quad 5c = 20 \quad\Longrightarrow\quad c = 4$$

Then $4 + 4t = 12$, so $t = 2$.

Check both sentences in words: 3 coffees and 2 teas is $12 + 4 = 16$ ✓, and 1 coffee and 4 teas is $4 + 8 = 12$ ✓

Why each wrong option is wrong:

• A, $2$ — the price of a tea. Define your letters in writing before you start; it is the cheapest insurance on the test.
• B, $\frac{14}{5}$ — divided all the money, $28, by all the drinks, 10, giving an average price per drink. That would be the answer only if coffee and tea cost the same, and the question exists because they do not.
• D, $28$ — added the two totals. That is the cost of everything on both orders, not the cost of one coffee.

Takeaway: Write down what each letter means before writing equations, and answer in those terms at the end. Then read both original sentences back with your numbers in them.

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Answer: A

Explanation

Two lines fail to have a crossing point exactly when they are parallel — same slope, different intercept. So put both into $y = mx + b$ form.

First equation:

$$6x - 2y = 10 \quad\Longrightarrow\quad -2y = -6x + 10 \quad\Longrightarrow\quad y = 3x - 5$$

Slope 3, intercept $-5$.

Second equation, keeping $k$:

$$9x + ky = 3 \quad\Longrightarrow\quad ky = -9x + 3 \quad\Longrightarrow\quad y = -\frac{9}{k}x + \frac{3}{k}$$

For the slopes to match:

$$-\frac{9}{k} = 3 \quad\Longrightarrow\quad -9 = 3k \quad\Longrightarrow\quad k = -3$$

Now confirm it is really "no solution" and not "infinitely many". With $k = -3$ the second line is $y = 3x - 1$. Same slope as $y = 3x - 5$, different intercept — so the lines are parallel and never meet ✓

That last check matters. Matching the slopes is only half the condition; if the intercepts had matched too, the two equations would describe the same line and the system would have infinitely many solutions instead.

Why each wrong option is wrong:

• B, $3$ — matched $\frac9k = 3$ and lost the minus sign that comes from moving $9x$ across.
• C, $- \frac{4}{3}$ — divided the wrong pair of coefficients.
• D, $-2$ — copied the $-2$ out of the first equation. The coefficients do have to be proportional, but $6 \to 9$ is a factor of $\frac32$, so $-2$ would have to become $-3$.

Takeaway: No solution means parallel: equal slopes, different intercepts. Infinitely many means the same line: equal slopes and equal intercepts. Always check the intercept after matching the slope — that is what separates the two cases.

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Answer: 7

Explanation

Infinitely many solutions means the two equations describe the same line — every point on it solves both.

Look at the left-hand sides. $4x + 6y$ is exactly twice $2x + 3y$. So the first equation is the second one doubled, and for them to be the same line the right-hand sides must be in that same ratio:

$$14 = 2c \quad\Longrightarrow\quad c = 7$$

Check by halving the first equation throughout:

$$4x + 6y = 14 \quad\Longrightarrow\quad 2x + 3y = 7$$

which is precisely the second equation with $c = 7$ ✓

Contrast this with Q12. There the coefficients were proportional but the constants were not, giving parallel lines and no solution. Here everything is in the same ratio, so it is one line drawn twice.

This is a grid-in. Enter 7.

Takeaway: Infinitely many solutions means one equation is a multiple of the other — coefficients and constant, all scaled by the same factor. Find the factor from the $x$ terms, then apply it to the constant.

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Answer: B

Explanation

The point where two lines cross is the point that lies on both of them. So its coordinates satisfy both equations at once — and that is exactly what "the solution of the system" means.

So $x = 4$ and $y = -2$ make both equations true. That is the whole content of an intersection point, and nothing else is forced.

To see that nothing else follows, here are two actual lines meeting at $(4, -2)$:

$$y = x - 6 \qquad\text{and}\qquad y = -x + 2$$

Check: at $x = 4$ the first gives $4 - 6 = -2$ ✓ and the second gives $-4 + 2 = -2$ ✓ They do cross there. Now test the other options against them.

Why each wrong option is wrong:

• A, "Both lines have a slope of 4" — the example lines have slopes 1 and $-1$. The 4 is an $x$-coordinate, not a slope. Two lines that cross must have different slopes, so they certainly cannot both be 4.
• C, "Both lines pass through the origin" — the example lines cross the $y$-axis at $-6$ and $2$. If both lines went through the origin they would cross there, not at $(4, -2)$.
• D, "The two lines have the same $y$-intercept" — again $-6$ and $2$. Two distinct lines meet at only one point; if they shared a $y$-intercept they would meet on the $y$-axis.

Takeaway: The solution of a system is the intersection point, and it means one thing only: those coordinates satisfy both equations. When a question asks what must be true, try to build an example that breaks each option — one counterexample is enough to eliminate it.

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Section 5 — Reading a Linear Model

The SAT dresses slope and intercept up as a story and asks what one of the numbers means. The answer is always about units.

In $y = mx + b$ describing a real situation:

• $b$ is the value at the start, when $x = 0$. Its units are the units of $y$.
• $m$ is how fast $y$ changes per unit of $x$. Its units are $y$-units per $x$-unit.

That unit test settles most of these questions on its own. If the model is dollars against hours, a rate must be dollars per hour, so any option measured in plain dollars or plain hours is out before you think about the story.

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Answer: C

Explanation

Set $w = 0$ — no weeks used yet:

$$C = 30 + 12(0) = 30$$

So $30 is owed before any weeks are counted. That is a joining fee.

A is what the 12 means: each extra week adds $12, since $C(w+1) - C(w) = 12$. B confuses a cost with a count — the 30 is in dollars. D would be $30 + 12(30) = 390$.

Takeaway: Substitute $x = 0$ to see what the constant means, and check the units. A dollar amount cannot be a number of weeks.

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Answer: A

Explanation

The number multiplied by $h$ is the rate. Between any hour and the next:

$$P(h+1) - P(h) = 80$$

So the population rises by 80 each hour, and it starts at 500.

B swaps the two numbers. C describes exponential growth, which this model is not — a linear model adds a fixed amount, it does not multiply. (After 80 hours the population is $500 + 6400 = 6900$, which is far more than double.) D reads the rate as the starting value.

Takeaway: Linear means adds the same amount each step. "Doubles every" is exponential and belongs to a different kind of model entirely — Chapter 5.

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Section 6 — Mixed Practice

Slopes, forms, parallels and systems, shuffled.

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Answer: D

Explanation

$$m = \frac{-3 - 7}{3 - (-2)} = \frac{-10}{5} = -2$$

The line falls from $y = 7$ to $y = -3$, so a negative slope is right ✓

A has the sign flipped. B is upside down. C added the $y$ values.

Takeaway: Subtracting a negative adds: $3 - (-2) = 5$. Then sanity-check the sign against whether the line rises or falls.

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Answer: B

Explanation

Rearrange: $-5y = -2x + 15$, so $y = \frac{2}{5}x - 3$ and $m = \frac25$.

Perpendicular means flip and negate: $-\frac52$. Check: $\frac25 \times -\frac52 = -1$ ✓

A flipped only. C is the parallel slope. D negated only.

Takeaway: Rearrange to $y = mx + b$ first, then do both halves of "negative reciprocal" and confirm the product is $-1$.

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Answer: 4

Explanation

The $y$ terms are $+2y$ and $-2y$ — opposite signs, so adding the equations cancels them:

$$(3x + 2y) + (5x - 2y) = 19 + 13 \quad\Longrightarrow\quad 8x = 32 \quad\Longrightarrow\quad x = 4$$

Then $3(4) + 2y = 19$ gives $2y = 7$, so $y = \frac72$.

Check both: $12 + 7 = 19$ ✓ and $20 - 7 = 13$ ✓

This is a grid-in. Enter 4.

Takeaway: Opposite signs on a variable, add the equations. Matching signs, subtract. Spotting which before you start saves the whole scaling step.

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Answer: C

Explanation

$$y = -\tfrac12 x + b \quad\Longrightarrow\quad 1 = -\tfrac12(6) + b = -3 + b \quad\Longrightarrow\quad b = 4$$

Check: at $x = 6$, $-\frac12(6) + 4 = -3 + 4 = 1$ ✓

A used the point's $y$-coordinate. B lost the minus on the slope. D added the coordinates together.

Takeaway: Substitute the point into $y = mx + b$ and solve for $b$. Carry the minus sign through the multiplication before doing anything else.

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Answer: 200

Explanation

"Charge the same" means the two expressions are equal — the point where the two lines cross:

$$40 + 0.20n = 10 + 0.35n$$

Gather the $n$ terms on the side that keeps them positive, by subtracting $0.20n$:

$$40 = 10 + 0.15n$$

Subtract 10:

$$30 = 0.15n \quad\Longrightarrow\quad n = \frac{30}{0.15} = 200$$

Check both quotes at 200 copies: firm A charges $40 + 40 = 80$, firm B charges $10 + 70 = 80$ ✓

Worth noticing which firm is cheaper on each side, because the test often asks that as a follow-up. Firm B starts cheaper (a $10 setup against $40) but charges more per copy, so it wins on small orders and loses on large ones. At 100 copies B costs $45 against A's $60; at 300 copies A costs $100 against B's $115.

This is a grid-in. Enter 200.

Takeaway: "The same cost", "break even", "when are they equal" all mean set the two expressions equal to each other. The one with the bigger rate always wins eventually; the crossing point is where the lead changes hands.

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Answer: C

Explanation

Nothing cancels as the equations stand, so scale them until something does. To clear $x$, make both $x$ terms $6x$ — multiply the first by 3 and the second by 2:

$$6x + 15y = 72 \qquad 6x - 4y = -4$$

Multiply every term, including the right-hand sides: $3 \times 24 = 72$ and $2 \times (-2) = -4$. Forgetting the right-hand side is the usual way this goes wrong.

Now subtract the second from the first. Bracket it, so the double negative is visible:

$$(6x + 15y) - (6x - 4y) = 72 - (-4)$$

$$19y = 76 \quad\Longrightarrow\quad y = 4$$

Then $2x + 5(4) = 24$ gives $2x = 4$, so $x = 2$.

Check both original equations: $2(2) + 5(4) = 4 + 20 = 24$ ✓ and $3(2) - 2(4) = 6 - 8 = -2$ ✓

Why each wrong option is wrong:

• A, $2$ — the value of $x$. Both 2 and 4 fall out of this question, and the last line named $y$.
• B, $76$ — reached $19y = 76$ and wrote 76 down. The unfinished last step.
• D, $-2$ — copied the constant from the second equation.

Takeaway: Scale until one variable's coefficients match, multiplying every term including the constant. Same sign, subtract; opposite signs, add. Then find the second variable and check both original equations.

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Answer: A

Explanation

Parallel means the same slope, 3, so the new line is $y = 3x + b$. Substitute $(2, 11)$:

$$11 = 3(2) + b = 6 + b \quad\Longrightarrow\quad b = 5$$

Check: at $x = 2$, $3(2) + 5 = 11$ ✓

B is the original line's intercept — parallel lines share a slope, never an intercept. C is the point's $y$-value. D added the 6 instead of subtracting it.

Takeaway: Copy the slope, substitute the point, solve for $b$. If you get the original intercept back, you have written down the same line.

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Answer: A

Explanation

The given slope is $-2$, which is $-\frac21$. Flip and negate: $m = \frac12$. Check: $-2 \times \frac12 = -1$ ✓

So the line is $y = \frac12 x + b$. Substitute $(4, 3)$:

$$3 = \tfrac12(4) + b = 2 + b \quad\Longrightarrow\quad b = 1$$

B used $-2$ as the new slope, so it built a parallel line. C gives the slope. D gives the original slope.

Takeaway: A whole number $-2$ is the fraction $-\frac21$; flipping it gives $\frac12$. Writing the whole number as a fraction first makes the flip obvious.

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Answer: 6

Explanation

On the $x$-axis, $y = 0$:

$$5x + 2(0) = 30 \quad\Longrightarrow\quad 5x = 30 \quad\Longrightarrow\quad x = 6$$

Check: $5(6) + 0 = 30$ ✓

Setting $x = 0$ instead would give $y = 15$ — a real point on the line, and the answer to the other question.

This is a grid-in. Enter 6.

Takeaway: $x$-axis means $y = 0$. Each axis is where the other coordinate is zero.

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Answer: B

Explanation

Let $a$ be adult tickets and $c$ child tickets. Two facts, two equations:

$$a + c = 250 \qquad 12a + 7c = 2400$$

Substitute $c = 250 - a$ into the money equation:

$$12a + 7(250 - a) = 2400$$

$$12a + 1750 - 7a = 2400 \quad\Longrightarrow\quad 5a = 650 \quad\Longrightarrow\quad a = 130$$

So 130 adult and 120 child tickets. Check: $130 + 120 = 250$ ✓ and $12(130) + 7(120) = 1560 + 840 = 2400$ ✓

A is the child count — both numbers come out, and the question named adults. C is the total. D assumed every ticket was an adult ticket.

Takeaway: Two unknowns need two equations: one counting the items, one counting the money. Then check both, and read which of the two the question wanted.

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Answer: A

Explanation

"Linear" means constant slope, so treat $f(2) = 5$ and $f(6) = 13$ as the points $(2, 5)$ and $(6, 13)$:

$$m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$$

So $f(x) = 2x + b$, and $5 = 2(2) + b$ gives $b = 1$. Since $f(0)$ is the $y$-intercept, $f(0) = 1$.

Check with the other point: $f(6) = 2(6) + 1 = 13$ ✓

There is a quicker way. Going from $x = 2$ down to $x = 0$ is two steps back, and each step is worth 2, so $f(0) = 5 - 2(2) = 1$.

B is $f(2)$. C is the slope. D averaged the two given outputs, which would be $f(4)$, not $f(0)$.

Takeaway: $f(0)$ is the $y$-intercept. With a linear function you can walk backwards from a known point in steps of the slope — often faster than solving for $b$.

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Answer: B

Explanation

There is nothing to solve. Substitute each point and see which one works — and with a calculator available this takes seconds.

$$\textbf{A } 3(0) - 4(3) = -12 \quad \text{no}$$ $$\textbf{B } 3(8) - 4(3) = 24 - 12 = 12 \quad ✓$$ $$\textbf{C } 3(4) - 4(3) = 12 - 12 = 0 \quad \text{no}$$ $$\textbf{D } 3(3) - 4(8) = 9 - 32 = -23 \quad \text{no}$$

Option D is worth a second look: it is B's coordinates swapped. The order of a coordinate pair is $(x, y)$, always, and a question offering both orders is checking that you know which is which.

Takeaway: "Which point lies on..." is a substitution question, not a solving question. Test the options; it is faster and it cannot go wrong.

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Answer: 14

Explanation

Put the two points into the slope formula and set it equal to 3:

$$\frac{k - 2}{5 - 1} = 3 \quad\Longrightarrow\quad \frac{k - 2}{4} = 3$$

Multiply both sides by 4:

$$k - 2 = 12 \quad\Longrightarrow\quad k = 14$$

Check: $\frac{14 - 2}{5 - 1} = \frac{12}{4} = 3$ ✓

Or walk it: from $x = 1$ to $x = 5$ is 4 steps, each worth 3, so $y$ climbs by 12, from 2 to 14.

This is a grid-in. Enter 14.

Takeaway: A missing coordinate with a known slope is one equation. Set the slope formula equal to the given slope and solve — or count the steps and multiply.

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Answer: A

Explanation

Rearrange, keeping $a$ as a letter:

$$ay = -4x + 8 \quad\Longrightarrow\quad y = -\frac{4}{a}x + \frac{8}{a}$$

The slope is $-\frac{4}{a}$, and it has to be $-2$:

$$-\frac{4}{a} = -2 \quad\Longrightarrow\quad 4 = 2a \quad\Longrightarrow\quad a = 2$$

Check by substituting back: $4x + 2y = 8$ becomes $y = -2x + 4$, slope $-2$ ✓

B copied the slope itself. C read the 8. D inverted the relationship, giving $a = \frac12$, which would make the slope $-8$.

Takeaway: Rearrange with the unknown constant still in place, then set the slope expression equal to the required value. Always substitute back — it costs one line and catches an inverted fraction immediately.

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Answer: C

Explanation

No solution means the lines are parallel but not identical — same slope, different intercept. Both equations are already in $y = mx + b$ form, so read them off:

Option	Slopes	Intercepts	Result
A	2 and 3	1 and 1	different slopes → they cross once
B	2 and 2	1 and 1	the same line → infinitely many
C	2 and 2	1 and $-4$	parallel, distinct → none
D	2 and $-2$	1 and 1	different slopes → they cross once

Only C has matching slopes with different intercepts.

Option B is the one to be careful with: identical equations look like they should be a special case of "no solution", and they are the opposite — every point on the line solves both.

Takeaway: Compare slopes first. Different slopes always cross. Same slope splits into two cases, and it is the intercept that decides which: same intercept means infinitely many, different means none.

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Answer: D

Explanation

Work out all four slopes and compare.

$$\textbf{A } m = -5 \qquad \textbf{B } m = \frac{8 - 0}{4 - 0} = 2$$ $$\textbf{C } 3x - y = 1 \Rightarrow y = 3x - 1,\; m = 3 \qquad \textbf{D } m = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4$$

The greatest is 4, option D.

The trap is option A. Its slope has the largest size, and $-5$ is the smallest of the four numbers. "Greatest slope" means greatest value, so a negative slope can never win — unless every option is negative.

Takeaway: Convert everything to a single number, then compare as numbers. "Greatest slope" is not "steepest"; a slope of $-5$ is steeper than 4 but far smaller.

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