Simultaneous Equations

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Notes

What are Simultaneous Equations?

Simultaneous equations involve **two unknowns** (usually x and y) and require **two equations** to find both values.
The solution is the pair of values that satisfy **both equations at the same time**.
They are called **linear** if there are no squared terms $(like x^{2}$ or $y^{2})$ .
Example: $3x + 2y = 11$ and 2x – $y = 5$ have solution $x = 3, y = 1$ .

Solving by Elimination

**Elimination** removes one variable by making the coefficients of x (or y) the same in both equations.
If the signs in front of the term are the **same**, **subtract** the equations.
If the signs are **different**, **add** the equations.
After eliminating one variable, solve the resulting equation for the other variable.
Substitute the found value into one original equation to find the second variable.
Always check your solutions in the **other** original equation.

Solving by Substitution

**Substitution** involves rearranging one equation to make x or y the subject $(e.g., y = 2x$ – 5).
Substitute this expression into the **other** equation.
Solve the resulting equation for the remaining variable.
Substitute back to find the other variable.
This method is an alternative to elimination.

Solving Graphically

Plot **both equations** on the same set of axes (use a table of values or rearrange to $y = mx + c)$ .
The **point of intersection** of the two lines gives the solution (x, y).
Example: 2x – $y = 3$ and $3x + y = 4$ intersect at (2, 1), so $x = 2, y = 1$ .
Graphical method is useful for checking answers but may be less precise.

Forming Simultaneous Equations from Word Problems

Introduce **two letters** (e.g., x and y) to represent the unknowns, stating what each stands for.
Translate the given information into **two equations**.
Example: '3 apples and 2 bananas cost $1.80' gives $3x + 2y = 180 (in cents)$ .
Solve the equations simultaneously, then **answer the question** in context (with units).
Sometimes you need to find a further value (e.g., product or total cost) after solving.

Worked Example (Elimination)

Solve: $5x + 2y = 11$ and 4x – $3y = 18$ .
Multiply first by $3: 15x + 6y = 33$ . Multiply second by 2: 8x – $6y = 36$ .
Add to eliminate $y: 23x = 69$ → $x = 3$ .
Substitute $x = 3$ into $5x + 2y = 11$ → $15 + 2y = 11$ → $2y =$ –4 → $y =$ –2.
Check in second: 4(3) – 3(–2) $= 12 + 6 = 18$ ✓. Solution: $x = 3, y =$ –2.

Worked Example (Forming Equations)

Customer 1: 6 bagels + 12 sausage rolls = £9 → $6b + 12s = 9$ .
Customer 2: 9 bagels + 10 sausage rolls = £12.30 → $9b + 10s = 12.3$ .
Eliminate b: multiply first by 3, second by 2 → $18b + 36s = 27$ and $18b + 20s = 24.6$ .
Subtract: $16s = 2.4$ → $s = 0.15$ . Substitute: $6b + 1.8 = 9$ → $b = 1.2$ .
Cost of 5 bagels and 15 sausage rolls $= 5\times 1.2 + 15\times 0.15 = 6 + 2.25 =$ £8.25.

Examiner Tips

**Always check** your final solutions satisfy both original equations.
Write both solutions together $(e.g., x = 3, y =$ –2) to avoid missing one.
Read the question carefully: sometimes you need to find something else (e.g., product, total cost).
Show all working clearly, especially when multiplying equations.

Graphical Solution of Simultaneous Equations

Practice questions

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1.Solve the simultaneous equations: $3x + 2y = 11$ and $2x - y = 5$ .
Easy
- A $x=3, y=1$
- B $x=1, y=3$
- C $x=2, y=2.5$
- D $x=4, y=-0.5$
2.Solve the simultaneous equations: $5x - 2y = 26$ and $7x + 6y = 10$ .
Easy
- A $x=4, y=-3$
- B $x=2, y=-8$
- C $x=6, y=2$
- D $x=-2, y=-18$
3.Solve the simultaneous equations: $6x - 3y = 12$ and $2x + 3y = 16$ .
Easy
- A $x=3.5, y=3$
- B $x=4, y=4$
- C $x=2, y=0$
- D $x=1, y=-2$
4.Solve the simultaneous equations: $5x + 4y = 10$ and $7x - 6y = 43$ .
Medium
- A $x=4, y=-2.5$
- B $x=5, y=-3.75$
- C $x=3, y=-1.25$
- D $x=2, y=0$
5.Esme buys x magazines at $2.45 each and y cards at $3.15 each. She spends $60.55 in total and buys 8 magazines. How many cards does she buy?
Medium
- A13
- B14
- C12
- D15
6.A shop sells pens and notebooks. A pen costs p cents, a notebook costs n cents. On Monday, 5 pens and 4 notebooks cost 450 cents. On Tuesday, 10 pens and 3 notebooks cost 525 cents. Find the cost of a pen.
Medium
- A30 cents
- B45 cents
- C60 cents
- D50 cents
7.Beindu buys 7 apples and 4 bananas for 85 cents, and 3 apples and 8 bananas for 93 cents. Find the cost of an apple (a cents) and a banana (b cents).
Medium
- A $a=7, b=9$
- B $a=8, b=7$
- C $a=9, b=6$
- D $a=6, b=10$
8.The Fraser family buys 6 adult tickets and 2 child tickets for $124. The Singh family buys 3 adult tickets and 5 child tickets for $100. Find the price of an adult ticket.
Hard
- A$18
- B$15
- C$20
- D$12

Summary

## Simultaneous Equations - Quick Revision ### Solving by Elimination 1. Make coefficients of one variable equal (multiply equations if needed). 2. If signs are **same**, subtract; if **different**, add. 3. Solve for the remaining variable. 4. Substitute back to find the other variable. 5. Check in both original equations. ### Solving by Substitution 1. Rearrange one equation to y = ... or x = ... 2. Substitute into the other equation. 3. Solve for the variable. 4. Substitute back to find the other. ### Solving Graphically - Plot both lines; the intersection point gives (x, y). ### Forming Equations from Word Problems - Define variables (e.g., x = price of an apple, y = price of a banana). - Write two equations from given information. - Solve simultaneously. - Answer in context with units. ### Key Points - Always show working. - Write final answer as x = ..., y = ... - Check solutions satisfy both equations. ### Example (Elimination) Solve: 5x + 2y = 11 (1) 4x - 3y = 18 (2) Multiply (1) by 3, (2) by 2: 15x + 6y = 33 8x - 6y = 36 Add: 23x = 69 → x = 3 Substitute into (1): 5(3)+2y=11 → 2y=-4 → y=-2 Answer: x=3, y=-2

Key terms

Simultaneous equations: Two equations with two unknowns (x and y) that are both true at the same time.
Elimination method: A method to solve simultaneous equations by adding or subtracting the equations to remove one variable.
Substitution method: A method to solve simultaneous equations by rearranging one equation and substituting it into the other.
Coefficient: The number in front of a variable (e.g., in 3x, the coefficient is 3).

Slides

What are Simultaneous Equations?

Simultaneous equations involve two unknowns (usually x and y) and two equations.
The solution is the pair of values that satisfy both equations at the same time.
For example, $3x + 2y = 11$ and $2x - y = 5$ have solution $x = 3, y = 1$ .

Which pair is a solution to $x + y = 5$ and $x - y = 1$ ?

$x=3, y=2$
$x=2, y=3$
$x=4, y=1$
$x=1, y=4$

Solving by Elimination

Elimination removes one variable by making coefficients the same.
If signs are the same, subtract the equations; if different, add them.
Solve the resulting equation, then substitute back to find the other variable.

To eliminate y from $3x + 2y = 11$ and $2x - y = 5$ , multiply the second equation $by ____ and$ then $____ the$ equations.

Solving by Substitution

Rearrange one equation to make x or y the subject.
Substitute this expression into the other equation.
Solve for one variable, then substitute back to find the other.

To solve $2x + y = 7$ and $x - y = 2$ by substitution, which rearrangement is best?

$y = 7 - 2x$
$y = 2x - 7$
$x = y + 2$
$x = 2 - y$

Solving Graphically

Plot both equations on the same axes.
The point where they intersect gives the solution (x, y).
This method is useful for checking your answer.

Which of these are true about solving simultaneous equations?

The solution must satisfy both equations.
Elimination always requires subtracting equations.
Graphically, the solution is the intersection point.

Worked Example (Elimination)

Solve: $5x + 2y = 11$ and $4x - 3y = 18$ .
Multiply first by $3: 15x + 6y = 33$ . Multiply second by $2: 8x - 6y = 36$ .
Add to eliminate $y: 23x = 69$ , so $x = 3$ . Substitute to get $y = -2$ .

After solving, check by substituting into the other equation: $4(3) - 3(-2) = ____ + ____ = 18$ .

Forming Simultaneous Equations

Introduce letters for unknowns (e.g., x for apples, y for bananas).
Write two equations from the given information.
Solve and then answer the question in context.

3 apples and 2 bananas cost $1.80. 5 apples and 1 banana cost $2.30. Which equations represent this?

$3x+2y=180, 5x+y=230$
$3x+2y=1.8, 5x+y=2.3$
$3x+2y=180, 5x+y=2.3$
$3x+2y=1.8, 5x+y=230$

Worked Example (Forming)

6 bagels and 12 sausage rolls cost £9. 9 bagels and 10 sausage rolls cost £12.30.
Let $b =$ bagel price, $s =$ sausage roll price. Equations: $6b+12s=9, 9b+10s=12.3$ .
Solve to get $b=1.2, s=0.15$ . Then 5 bagels and 15 rolls cost 5(1.2)+15(0.15)=£8.25.

The cost of 5 bagels and 15 sausage rolls is £____.

Checking Your Answer

Always substitute your solution back into both original equations.
Write both x and y together at the end.
Make sure you answer the question (e.g., find the cost, not just x and y).

After solving, you get $x=4, y=2$ . Which check shows it's correct for $3x+2y=16$ ?

$3(4)+2(2)=12+4=16$
$3(4)+2(2)=12+2=14$
$3(4)+2(2)=7+4=11$
$3(4)+2(2)=8+4=12$

Flashcards

What are linear simultaneous equations?

Two equations with two unknowns (x and y) that are solved together because the same values satisfy both equations at the same time.

Solve: $3x - 8y = 22$ and $x + 4y = 4$

$x = 6, y = -0.5 (or y = -\frac{1}{2})$

Solve: $3x - y = 22$ and $x + 2y = 5$

$x = 7, y = -1$

Solve: $5x - 2y = 26$ and $7x + 6y = 10$

$x = 4, y = -3$

Solve: $6x - 3y = 12$ and $2x + 3y = 16$

$x = 3.5, y = 3$

Solve: $5x - 2y = 44$ and $2x + 3y = 10$

$x = 8, y = -2$

Solve: $5x + 4y = 10$ and $7x - 6y = 43$

$x = 4, y = -2.5$

Find the intersection point of $y = 3x + 13$ and $y = 7x - 3$ .

$x = 4, y = 25$

Solve: $8x - 3y = 19$ and $4x + 5y = 3$

$x = 2, y = -1$

Solve: $5x + 2y = 11$ and $4x - 3y = 18$

$x = 3, y = -2$

What is the elimination method for solving simultaneous equations?

Make the coefficients of one variable the same in both equations, then add or subtract to eliminate that variable and solve for the other.

When eliminating, when do you add equations?

When the signs in front of the term you want to eliminate are different.

When eliminating, when do you subtract equations?

When the signs in front of the term you want to eliminate are the same.

How do you solve simultaneous equations by substitution?

Rearrange one equation to make x or y the subject, then substitute that expression into the other equation.

How do you solve simultaneous equations graphically?

Plot both lines on the same axes; the intersection point gives the solution (x, y).

Esme buys x magazines at $2.45 each and y cards at $3.15 each. Total cost $60.55, she buys 8 magazines. How many cards?

13 cards $(2.45\times 8 + 3.15y = 60.55$ → $y = 13)$

5 pens and 4 notebooks cost 450 cents. 10 pens and 3 notebooks cost 525 cents. Find cost of pen and notebook.

$Pen = 30$ cents, Notebook $= 75$ cents

7 apples and 4 bananas cost 85 cents. 3 apples and 8 bananas cost 93 cents. Find cost of apple and banana.

Apple $= 7$ cents, Banana $= 9$ cents

6 adult tickets and 2 child tickets cost $124. 3 adult and 5 child cost $100. Find adult and child ticket prices.

Adult = $15, Child = $17

2 plants and 4 bushes cost $42. 7 plants and 9 bushes cost $107. Find p and b.

$p = 5, b = 8$

Three times Alphonse's age equals five times Beatrice's age. Twice Beatrice's age is 4 years more than Alphonse's age. Find their ages.

Alphonse $= 20$ , Beatrice $= 12$

Two numbers sum to -6 and differ by 62. Find the numbers.

28 and -34

Minibus hire: 3 hours, 10 passengers costs $61; 5 hours, 8 passengers costs $80. Find h and p.

$h = 12, p = 2.5$

3 cucumbers and 5 tomatoes cost £14.50. 5 cucumbers and 4 tomatoes cost £15.50. Find cost of each.

Cucumber = £1.50, Tomato = £2.00

6 bagels and 12 sausage rolls cost £9. 9 bagels and 10 sausage rolls cost £12.30. Find cost of 5 bagels and 15 sausage rolls.

£8.25

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