Foundation algebra and graph clinic

(a) $x + 14$

1. Expand; the $-2$ multiplies both terms.
   $$3x + 12 - 2x + 2$$
2. Collect.
   $$x + 14$$

• M1 Expanding both brackets correctly: $3x + 12$ and $-2x + 2$.
• A1 The correct answer, $x + 14$.

(b) $x = 6$

1. Using part (a): $x + 14 = 20$, so $x = 6$.

• M1 Writing $x + 14 = 20$ (or their simplified expression $= 20$).
• A1 The correct answer, $x = 6$.

(a) $y = 2x + 1$

1. Gradient: $\frac{7 - 1}{3 - 0} = 2$. The line crosses the $y$-axis at 1.
2. So $y = 2x + 1$.

• M1 Finding the gradient, 2.
• A1 The correct answer, $y = 2x + 1$.

(b) Yes: $2 \times 5 + 1 = 11$.

1. When $x = 5$, $y = 2 \times 5 + 1 = 11$. It matches, so the point is on the line.

• M1 Substituting $x = 5$.
• C1 "Yes", with $y = 11$ shown.

(a) $36-6x$

1. Distribute minus 3 to both terms inside the bracket.
   $$12-6x+24$$
2. Therefore $36-6x$.

• M1 Distribute minus 3 to both terms inside the bracket.
• A1 Correct answer: $36-6x$

(b) $6$

1. Set the simplified expression equal to zero and isolate x.
   $$6x=36$$
2. Therefore $6$.

• M1 Set the simplified expression equal to zero and isolate x.
• A1 Correct answer: $6$

(a) $12$

1. Multiply every term by 6.
   $$2(x-18)+3(x+9)=51$$
2. Collect terms and isolate x.
   $$5x=60$$
3. Therefore $12$.

• M1 Multiply every term by 6.
• M1 Collect terms and isolate x.
• A1 Correct answer: $12$

(b) $23$

1. Use the solved value in the requested expression.
   $$2\times 12-1$$
2. Therefore $23$.

• M1 Use the solved value in the requested expression.
• A1 Correct answer: $23$

(a) −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

1. Subtract 1 from all three expressions and divide by 2.
   $$-11/2<x\le 12$$
2. Therefore −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

• M1 Subtract 1 from all three expressions and divide by 2.
• A1 Correct answer: −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

(b) $63$

1. Add the finite arithmetic list using its average and count.
   $$18\times (-5+12)/2$$
2. Therefore $63$.

• M1 Add the finite arithmetic list using its average and count.
• A1 Correct answer: $63$

(a) $y=10x-8$

1. Calculate the gradient as rise divided by run.
   $$\frac{152-122}{3}$$
2. Substitute a known point to find the intercept.
   $$122-10\times 13$$
3. Therefore $y=10x-8$.

• M1 Calculate the gradient as rise divided by run.
• M1 Substitute a known point to find the intercept.
• A1 Correct answer: $y=10x-8$

(b) $y=10x-3$

1. Parallel lines have equal gradients.
   $$y=10x-3$$
2. Therefore $y=10x-3$.

• M1 Parallel lines have equal gradients.
• A1 Correct answer: $y=10x-3$

(a) $16$ cm

1. Equate the product of the lengths to the area.
   $$x(x+14)=480$$
2. Factorise the quadratic after moving the area to the left.
   $$(x-16)(x+30)=0$$
3. Therefore $16$ cm.

• P1 Equate the product of the lengths to the area.
• P1 Factorise the quadratic after moving the area to the left.
• A1 Correct answer: $16$ cm

(b) The width is a positive length, so the negative root is rejected.

1. The width is a positive length, so the negative root is rejected.

• C1 Correct conclusion with supporting reasoning: The width is a positive length, so the negative root is rejected.

(a) $x = 4$, $y = 3$

1. Add the equations.
   $$3x = 12 \Rightarrow x = 4$$
2. Substitute.
   $$4 - y = 1 \Rightarrow y = 3$$

• M1 Eliminating $y$: $3x = 12$.
• A1 $x = 4$ and $y = 3$.

(b) $x = 3$ or $x = -4$

1. $(x + 4)(x - 3) = 0$, so $x = -4$ or $x = 3$.

• M1 Factorising to $(x + 4)(x - 3)$.
• A1 $x = 3$ and $x = -4$.