Solving Linear Equations GCSE Questions

Answer $x = 5$

1. $\displaystyle 3x = 15$Subtract $5$ from both sides
2. $\displaystyle x = 5$Divide both sides by $3$

• M1 $3x = 15$
• A1 $x = 5$

Answer $x = 20$

1. $\displaystyle \frac{x}{4} = 5$Add $3$ to both sides
2. $\displaystyle x = 20$Multiply both sides by $4$

• M1 $\dfrac{x}{4} = 5$
• A1 $x = 20$

Answer $x = 9$

1. $\displaystyle x - 2 = 7$Divide both sides by $5$ (or expand to $5x - 10 = 35$)
2. $\displaystyle x = 9$

• M1 $5x - 10 = 35$ or $x - 2 = 7$
• A1 $x = 9$

Answer $x = 5$

1. $\displaystyle 4x - 4 = 16$Subtract $3x$ from both sides
2. $\displaystyle 4x = 20$Add $4$ to both sides
3. $\displaystyle x = 5$

• M1 Collects the $x$ terms: $4x - 4 = 16$ or $7x - 3x = 16 + 4$
• M1 $4x = 20$
• A1 $x = 5$

Answer $x = \tfrac{1}{2}$

1. $\displaystyle 6x + 2 = 5$Expand the bracket
2. $\displaystyle 6x = 3$
3. $\displaystyle x = \tfrac{3}{6} = \tfrac{1}{2}$An answer can be a fraction

• M1 $6x + 2 = 5$
• A1 $x = \tfrac{1}{2}$ oe

Answer $x = -11$

1. $\displaystyle 4x + 12 = 2x - 10$Expand both brackets
2. $\displaystyle 2x + 12 = -10$Subtract $2x$ from both sides
3. $\displaystyle 2x = -22$
4. $\displaystyle x = -11$

• M1 $4x + 12 = 2x - 10$
• M1 $2x = -22$
• A1 $x = -11$

Answer $x = 7$

1. $\displaystyle x + 5 = 3(x - 3)$Multiply both sides by $3$
2. $\displaystyle x + 5 = 3x - 9$
3. $\displaystyle 14 = 2x$Subtract $x$ and add $9$ on both sides
4. $\displaystyle x = 7$

• M1 $x + 5 = 3(x - 3)$
• M1 $x + 5 = 3x - 9$ and collects terms, e.g. $14 = 2x$
• A1 $x = 7$

Answer $x = 4$

1. $\displaystyle 12 = 5x - 8$Add $3x$ to both sides, so the $x$ term stays positive
2. $\displaystyle 20 = 5x$
3. $\displaystyle x = 4$

• M1 Collects the $x$ terms on one side: e.g. $12 = 5x - 8$
• M1 $20 = 5x$
• A1 $x = 4$

Answer $x = 8$

1. $\displaystyle 2x - 1 = 15$Multiply both sides by $5$
2. $\displaystyle 2x = 16$
3. $\displaystyle x = 8$

• M1 $2x - 1 = 15$
• A1 $x = 8$

Answer $x = 6$

1. $\displaystyle 5(x + 2) + 4(x - 1) = 60$Multiply every term, including the $3$, by $20$
2. $\displaystyle 5x + 10 + 4x - 4 = 60$
3. $\displaystyle 9x + 6 = 60$
4. $\displaystyle 9x = 54 \;\Rightarrow\; x = 6$

• M1 Multiplies every term by $20$: $5(x + 2) + 4(x - 1) = 60$
• M1 $9x + 6 = 60$
• A1 $x = 6$

Answer $x = 5$

1. $\displaystyle 3(3x - 1) - 2(x + 4) = 24$Multiply every term by $6$
2. $\displaystyle 9x - 3 - 2x - 8 = 24$Careful: $-2(x + 4) = -2x - 8$
3. $\displaystyle 7x - 11 = 24$
4. $\displaystyle 7x = 35 \;\Rightarrow\; x = 5$

• M1 Multiplies every term by $6$: $3(3x - 1) - 2(x + 4) = 24$
• M1 $7x - 11 = 24$
• A1 $x = 5$

Answer $80^\circ$

1. $\displaystyle x + (2x + 10) + (3x - 40) = 180$Angles in a triangle add up to $180^\circ$
2. $\displaystyle 6x - 30 = 180$
3. $\displaystyle 6x = 210 \;\Rightarrow\; x = 35$
4. $\displaystyle 35^\circ,\quad 2(35) + 10 = 80^\circ,\quad 3(35) - 40 = 65^\circ$Work out all three angles
5. $\displaystyle \text{Largest angle} = 80^\circ$The question asks for the angle, not $x$

• M1 $x + 2x + 10 + 3x - 40 = 180$
• M1 $6x - 30 = 180$
• A1 $x = 35$
• A1 $80^\circ$

Answer $x = 5$

1. $\displaystyle 2(2x + 3) + 2(x - 1) = 34$Perimeter: two lengths and two widths
2. $\displaystyle 4x + 6 + 2x - 2 = 34$
3. $\displaystyle 6x + 4 = 34$
4. $\displaystyle 6x = 30 \;\Rightarrow\; x = 5$

• M1 $2(2x + 3) + 2(x - 1) = 34$ oe
• M1 $6x + 4 = 34$
• A1 $x = 5$

Answer $52\ \text{cm}^2$

1. $\displaystyle \text{length} = 2(5) + 3 = 13,\quad \text{width} = 5 - 1 = 4$
2. $\displaystyle \text{area} = 13 \times 4 = 52\ \text{cm}^2$

• M1 Length $13$ and width $4$ — ft their $x$
• A1 $52\ \text{cm}^2$

Answer $5$ years old

1. $\displaystyle \text{Ben} = x,\quad \text{Ali} = 3x$Choose a letter for the unknown
2. $\displaystyle 3x + 5 = 2(x + 5)$In $5$ years both are $5$ years older
3. $\displaystyle 3x + 5 = 2x + 10$
4. $\displaystyle x = 5$Check: Ali is $15$; in $5$ years, $20 = 2 \times 10$

• M1 $3x + 5 = 2(x + 5)$, where Ben is $x$ now
• M1 $3x + 5 = 2x + 10$
• A1 $5$

Answer Moving $-7$ to the other side makes it $+7$, not $-7$. $2x = 16$, so $x = 8$.

1. $\displaystyle 4x - 2x = 9 + 7$Adding $7$ to both sides gives $+7$ on the right
2. $\displaystyle 2x = 16$
3. $\displaystyle x = 8$

• B1 Explains that the $-7$ becomes $+7$ on the other side (she should add $7$)
• B1 $x = 8$

Answer $x = 6.5$

1. $\displaystyle 3(x + 1) = 5(x - 2)$Multiply both sides by $(x - 2)(x + 1)$: cross-multiply
2. $\displaystyle 3x + 3 = 5x - 10$
3. $\displaystyle 13 = 2x$
4. $\displaystyle x = 6.5$

• M1 $3(x + 1) = 5(x - 2)$
• M1 $3x + 3 = 5x - 10$
• A1 $x = 6.5$ oe

Answer $x = 3$

1. $\displaystyle x^2 + 2x + 1 = x^2 - 6x + 9 + 16$Expand both brackets
2. $\displaystyle 2x + 1 = -6x + 25$The $x^2$ terms cancel, so it is linear
3. $\displaystyle 8x = 24$
4. $\displaystyle x = 3$

• M1 Expands both squares: $x^2 + 2x + 1$ and $x^2 - 6x + 9$
• M1 $8x = 24$ or $2x + 1 = -6x + 25$
• A1 $x = 3$

Answer $x = 7$

1. $\displaystyle \frac{4 + 7 + x + 2x}{4} = 8$Mean $=$ total $\div$ how many
2. $\displaystyle 11 + 3x = 32$Multiply both sides by $4$
3. $\displaystyle 3x = 21 \;\Rightarrow\; x = 7$

• M1 $\dfrac{4 + 7 + x + 2x}{4} = 8$
• M1 $11 + 3x = 32$
• A1 $x = 7$

Answer $10.5$ miles

1. $\displaystyle 3.50 + 1.20m = 16.10$Let $m$ be the number of miles
2. $\displaystyle 1.20m = 12.60$
3. $\displaystyle m = 12.60 \div 1.20 = 10.5$

• M1 $3.50 + 1.20m = 16.10$
• M1 $1.20m = 12.60$
• A1 $10.5$

Answer $37$ and $39$

1. $\displaystyle x + (x + 2) = 76$Consecutive odd numbers are $2$ apart
2. $\displaystyle 2x + 2 = 76 \;\Rightarrow\; 2x = 74$
3. $\displaystyle x = 37$The numbers are $37$ and $39$

• M1 $x + (x + 2) = 76$
• M1 $2x = 74$
• A1 $37$ and $39$

Answer Expanding gives $6x + 2 - 6x + 3 = 4$, so $5 = 4$, which is false. No value of $x$ works.

1. $\displaystyle 6x + 2 - 6x + 3 = 4$Careful: $-3(2x - 1) = -6x + 3$
2. $\displaystyle 5 = 4$The $x$ terms cancel and leave a false statement
3. $\displaystyle \text{No value of } x \text{ makes it true}$

• M1 Expands correctly: $6x + 2 - 6x + 3$
• A1 $5 = 4$ (or $5 \ne 4$) with a conclusion that there is no solution

Answer $x = \tfrac{1}{3}$

1. $\displaystyle (x + 3)(x - 1) = (x + 1)(x - 2)$Cross-multiply
2. $\displaystyle x^2 + 2x - 3 = x^2 - x - 2$Expand both sides
3. $\displaystyle 2x - 3 = -x - 2$The $x^2$ terms cancel
4. $\displaystyle 3x = 1 \;\Rightarrow\; x = \tfrac{1}{3}$

• M1 $(x + 3)(x - 1) = (x + 1)(x - 2)$
• M1 $x^2 + 2x - 3 = x^2 - x - 2$
• M1 $3x = 1$
• A1 $x = \tfrac{1}{3}$

Answer $k = \tfrac{7}{4}$

1. $\displaystyle \frac{14 + k}{3} = 7 - k$Put $x = 7$ into the equation: now $k$ is the unknown
2. $\displaystyle 14 + k = 21 - 3k$Multiply both sides by $3$
3. $\displaystyle 4k = 7$
4. $\displaystyle k = \tfrac{7}{4}$

• M1 Substitutes $x = 7$: $\dfrac{14 + k}{3} = 7 - k$
• M1 $14 + k = 21 - 3k$
• A1 $k = \tfrac{7}{4}$ oe

Answer $7.5$ km

1. $\displaystyle \frac{d}{15} + \frac{d}{5} = 2$Time $=$ distance $\div$ speed, for each journey
2. $\displaystyle d + 3d = 30$Multiply every term by $15$
3. $\displaystyle 4d = 30 \;\Rightarrow\; d = 7.5$

• M1 $\dfrac{d}{15} + \dfrac{d}{5} = 2$
• M1 $d + 3d = 30$ or $\dfrac{4d}{15} = 2$
• A1 $7.5$ km