Linear formulas and equations

Answer: $92.5$

Substitute $d = 31\mathclose{:}$ $C = 15 + 2.5 \times 31 = 15 + 77.5 = 92.5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Left out the fixed amount → $77.5$
• Added the two numbers before multiplying → $542.5$

Answer: $3$

On the $x$-axis $y = 0\mathclose{,}$ so $5 x = 15$ and $x = 3\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the other intercept → $5$
• Divided the coefficient by the constant → $\dfrac{1}{3}$

Answer: $y = - \frac{3 x}{2} + 1$

The slope is $m = \dfrac{10 - 4}{-6 - (-2)} = -\dfrac{6}{4} = -\dfrac{3}{2}\mathclose{.}$ The point $(-2, 4)$ lies on the line, so $4 = -\dfrac{3}{2} \cdot (-2) + c\mathclose{,}$ which gives $c = 1\mathclose{.}$ The line is $y = - \frac{3 x}{2} + 1\mathclose{.}$

Common mistakes, and the answer each one gives:

• Inverted the slope (run over rise) → $y = - \frac{2 x}{3} + \frac{8}{3}$
• Found the slope but left out the intercept → $y = - \frac{3 x}{2}$
• Got the sign of the intercept wrong → $y = - \frac{3 x}{2} - 1$

Answer: $2$

The point is on the graph, so its coordinates fit the formula: $12 = p^2 + 8\mathclose{,}$ so $p^2 = 4$ and, as $p > 0\mathclose{,}$ $p = 2\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot to take the square root → $4$
• Gave the negative root → $-2$

Answer: $(-5, 4)$

Both give the same $y$ there: $- 3 x - 11 = - 4 x - 16\mathclose{,}$ so $x = -5\mathclose{.}$ Then $y = 4\mathclose{:}$ the point is $(-5, 4)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Moved the $x$ term across without changing its sign → $(\dfrac{5}{7}, -\dfrac{92}{7})$
• Found $x$ and gave it as both coordinates → $(-5, -5)$

Answer: $-17$

Expand the bracket: $4 x + 28 = 2 x - 6\mathclose{.}$ Put the $x$ terms on the left and the numbers on the right, changing the sign of each term that crosses over: $2 x = -34\mathclose{,}$ so $x = -17\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $- \frac{13}{2}$
• Moved the number across without changing its sign → $11$

Answer: $8$

Multiply both sides by $28$ to clear the denominators: $7\left(x - 4\right) = 4\left(x - 1\right)\mathclose{.}$ Expand: $7 x - 28 = 4 x - 4\mathclose{.}$ Collect: $3 x = 24\mathclose{,}$ so $x = 8\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied out $4\left(x - 1\right)$ as $4 x - 1\mathclose{,}$ multiplying only the $x$ → $9$
• Cross-multiplied the wrong way, writing $4\left(x - 4\right) = 7\left(x - 1\right)$ → $-3$

Answer: $x > 6$

Collect the $x$ terms on the left and the numbers on the right: $- 6 x < -36\mathclose{.}$ Divide by $-6$ — a NEGATIVE number, so the inequality sign turns round: $x > 6\mathclose{.}$

Common mistakes, and the answer each one gives:

• Did not reverse the inequality when dividing by a negative number → $x < 6$
• Moved the number across without changing its sign → $x > \frac{10}{3}$

Answer: $12$

Put $K = 35$ and solve for $m\mathclose{:}$ $35 - 5 = 2.5m\mathclose{,}$ so $m = 30 \div 2.5 = 12\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided by the rate before taking the fixed amount off → $9$
• Added the fixed amount instead of subtracting it → $16$

Answer: $y = 5 x - 20$

Parallel lines have the SAME slope, $5\mathclose{.}$ Through $(4, 0)\mathclose{:}$ $y = 5\left(x - 4\right)\mathclose{,}$ which is $y = 5 x - 20\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used the perpendicular slope → $y = - \frac{x}{5} + \frac{4}{5}$
• Kept the intercept of the given line → $y = 5 x + 4$

Answer: $-2$

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{8 - 2}{-7 - (-4)} = -\dfrac{6}{3} = -2\mathclose{.}$

Common mistakes, and the answer each one gives:

• Inverted the fraction (run over rise) → $- \frac{1}{2}$
• Subtracted the coordinates in opposite orders → $2$

Answer: $t = \dfrac{v - u}{a}$

Undo what is done to $t$ in reverse order, the same to both sides: take $u$ off both sides, $v - u = at\mathclose{,}$ then divide by $a\mathclose{.}$ So $t = \dfrac{v - u}{a}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added instead of subtracting $u$ → $t = \dfrac{v + u}{a}$
• Divided only $u$ by $a$ → $t = v - \dfrac{u}{a}$

Answer: $y = x + \frac{6}{5}$

Move the $x$ term across, changing its sign: $5y = 5 x + 6\mathclose{.}$ Then divide EVERY term by $5\mathclose{:}$ $y = x + \frac{6}{5}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided only the number by the coefficient of $y$ → $y = 5 x + \frac{6}{5}$
• Lost the sign of the $x$ term moving it across → $y = - x + \frac{6}{5}$

Answer: $14$

Substitute $m = 40\mathclose{:}$ $K = 4 + 0.25 \times 40 = 4 + 10 = 14\mathclose{.}$

Common mistakes, and the answer each one gives:

• Left out the fixed amount → $10$
• Added the two numbers before multiplying → $170$

Answer: $y = - \frac{x}{2} - 3$

The slope is $m = \dfrac{-9 - (-7)}{12 - 8} = -\dfrac{2}{4} = -\dfrac{1}{2}\mathclose{.}$ The point $(8, -7)$ lies on the line, so $-7 = -\dfrac{1}{2} \cdot 8 + c\mathclose{,}$ which gives $c = -3\mathclose{.}$ The line is $y = - \frac{x}{2} - 3\mathclose{.}$

Common mistakes, and the answer each one gives:

• Inverted the slope (run over rise) → $y = - 2 x + 9$
• Found the slope but left out the intercept → $y = - \frac{x}{2}$
• Got the sign of the intercept wrong → $y = - \frac{x}{2} + 3$

Answer: $-6$

On the $y$-axis $x = 0\mathclose{,}$ so $2 y = -12$ and $y = -6\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the other intercept → $-4$
• Divided the coefficient by the constant → $-\dfrac{1}{6}$

Answer: $53$

Expand the bracket: $5 x - 45 = 4 x + 8\mathclose{.}$ Put the $x$ terms on the left and the numbers on the right, changing the sign of each term that crosses over: $x = 53\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $17$
• Moved the number across without changing its sign → $-37$

Answer: $6$

The point is on the graph, so its coordinates fit the formula: $27 = p^2 - 9\mathclose{,}$ so $p^2 = 36$ and, as $p > 0\mathclose{,}$ $p = 6\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot to take the square root → $36$
• Gave the negative root → $-6$

Answer: $(-3, 6)$

Both give the same $y$ there: $- 2 x = x + 9\mathclose{,}$ so $- 3 x = 9$ and $x = -3\mathclose{.}$ Then $y = 6\mathclose{:}$ the point is $(-3, 6)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Moved the $x$ term across without changing its sign → $(-9, 18)$
• Found $x$ and gave it as both coordinates → $(-3, -3)$

Answer: $-\dfrac{1}{3}$

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-5 - (-4)}{4 - 1} = -\dfrac{1}{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Inverted the fraction (run over rise) → $-3$
• Subtracted the coordinates in opposite orders → $\frac{1}{3}$

Answer: $36$

Put $C = 95$ and solve for $d\mathclose{:}$ $95 - 5 = 2.5d\mathclose{,}$ so $d = 90 \div 2.5 = 36\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided by the rate before taking the fixed amount off → $33$
• Added the fixed amount instead of subtracting it → $40$

Answer: $y = - 4 x + 1$

Parallel lines have the SAME slope, $-4\mathclose{.}$ Through $(1, -3)\mathclose{:}$ $y = -4\left(x - 1\right) - 3\mathclose{,}$ which is $y = - 4 x + 1\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used the perpendicular slope → $y = \frac{x}{4} - \frac{13}{4}$
• Kept the intercept of the given line → $y = - 4 x - 8$

Answer: $-93$

Multiply both sides by $30$ to clear the denominators: $6\left(x + 8\right) = 5\left(x - 9\right)\mathclose{.}$ Expand: $6 x + 48 = 5 x - 45\mathclose{.}$ Collect: $x = -93\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied out $5\left(x - 9\right)$ as $5 x - 9\mathclose{,}$ multiplying only the $x$ → $-57$
• Cross-multiplied the wrong way, writing $5\left(x + 8\right) = 6\left(x - 9\right)$ → $94$

Answer: $x > 5$

Collect the $x$ terms on the left and the numbers on the right: $2 x > 10\mathclose{.}$ Divide by $2\mathclose{,}$ a positive number, so the sign stays: $x > 5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot to divide the number side by $2$ → $x > 10$
• Moved the number across without changing its sign → $x > 7$

Answer: $w = \dfrac{P - 2l}{2}$

Undo what is done to $w$ in reverse order, the same to both sides: take $2l$ off both sides, $P - 2l = 2w\mathclose{,}$ then divide by 2. So $w = \dfrac{P - 2l}{2}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided only $2l$ by 2 → $w = P - l$
• Took off $l$ instead of $2l$ → $w = \dfrac{P - l}{2}$

Answer: $y = \frac{2 x}{5} + \frac{9}{5}$

Move the $x$ term across, changing its sign: $5y = 2 x + 9\mathclose{.}$ Then divide EVERY term by $5\mathclose{:}$ $y = \frac{2 x}{5} + \frac{9}{5}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided only the number by the coefficient of $y$ → $y = 2 x + \frac{9}{5}$
• Lost the sign of the $x$ term moving it across → $y = - \frac{2 x}{5} + \frac{9}{5}$

Answer: $-\dfrac{25}{3}$

Expand the bracket: $5 x + 25 = 2 x\mathclose{.}$ Put the $x$ terms on the left and the numbers on the right, changing the sign of each term that crosses over: $3 x = -25\mathclose{,}$ so $x = -\dfrac{25}{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $- \frac{5}{3}$
• Moved the number across without changing its sign → $\frac{25}{3}$

Answer: $y = 2 x + 5$

The slope is $m = \dfrac{-13 - (-7)}{-9 - (-6)} = \dfrac{6}{3} = 2\mathclose{.}$ The point $(-6, -7)$ lies on the line, so $-7 = 2 \cdot (-6) + c\mathclose{,}$ which gives $c = 5\mathclose{.}$ The line is $y = 2 x + 5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Inverted the slope (run over rise) → $y = \frac{x}{2} - 4$
• Found the slope but left out the intercept → $y = 2 x$
• Got the sign of the intercept wrong → $y = 2 x - 5$

Answer: $5$

Substitute $t = 5\mathclose{:}$ $h = 20 - 3 \times 5 = 20 - 15 = 5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Left out the fixed amount → $-15$
• Added the term instead of subtracting it → $35$

Answer: $4$

On the $x$-axis $y = 0\mathclose{,}$ so $x = 4\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the other intercept → $2$
• Divided the coefficient by the constant → $\dfrac{1}{4}$