Solving Simultaneous Equations

Finding where two equations meet

What are Simultaneous Equations?

Simultaneous equations are two or more equations that share the same variables and must be solved together. The solution is the set of values that satisfies ALL equations at the same time.

For two linear equations, the solution represents the point where the two lines intersect on a graph.

Methods of Solving Simultaneous Equations

There are three main methods for solving simultaneous equations:

1. Elimination Method

Add or subtract equations to eliminate one variable, then solve for the remaining variable.

Best when: Coefficients are easy to match

2. Substitution Method

Express one variable in terms of the other, then substitute into the second equation.

Best when: One equation has a coefficient of 1

Method 1: Elimination

Example 1: Elimination by Addition

Solve the simultaneous equations:

\[2x + 3y = 13 \quad \text{...(1)}\]

\[5x - 3y = 8 \quad \text{...(2)}\]

1 Add the equations (since the y-coefficients are opposite):

\[(2x + 3y) + (5x - 3y) = 13 + 8\]

\[7x = 21\]

2 Solve for x:

\[x = \frac{21}{7} = 3\]

3 Substitute x = 3 into equation (1):

\[2(3) + 3y = 13\]

\[6 + 3y = 13\]

\[3y = 7\]

\[y = \frac{7}{3}\]

4 Solution: $x = 3$ and $y = \frac{7}{3}$

Example 2: Elimination by Making Coefficients Equal

Solve:

\[3x + 2y = 12 \quad \text{...(1)}\]

\[2x + 5y = 19 \quad \text{...(2)}\]

1 Make the x-coefficients equal by multiplying:

Equation (1) × 2: $6x + 4y = 24$ ...(3)

Equation (2) × 3: $6x + 15y = 57$ ...(4)

2 Subtract equation (3) from (4):

\[(6x + 15y) - (6x + 4y) = 57 - 24\]

\[11y = 33\]

\[y = 3\]

3 Substitute y = 3 into equation (1):

\[3x + 2(3) = 12\]

\[3x + 6 = 12\]

\[3x = 6\]

\[x = 2\]

4 Solution: $x = 2$ and $y = 3$

Method 2: Substitution

Example 3: Substitution Method

Solve:

\[y = 2x + 1 \quad \text{...(1)}\]

\[3x + 2y = 12 \quad \text{...(2)}\]

1 Substitute equation (1) into equation (2):

\[3x + 2(2x + 1) = 12\]

2 Expand and solve:

\[3x + 4x + 2 = 12\]

\[7x + 2 = 12\]

\[7x = 10\]

\[x = \frac{10}{7}\]

3 Find y using equation (1):

\[y = 2\left(\frac{10}{7}\right) + 1 = \frac{20}{7} + \frac{7}{7} = \frac{27}{7}\]

4 Solution: $x = \frac{10}{7}$ and $y = \frac{27}{7}$

Interactive Simultaneous Equations Solver

Graphical Solver

Enter coefficients for two linear equations in the form $ax + by = c$

Equation 1

x + y =

Equation 2

x + y =

Solution

Enter coefficients and click "Solve & Graph"

Word Problems

Example 4: Age Problem

Problem: John is 4 years older than Mary. In 6 years, the sum of their ages will be 40. Find their current ages.

1 Define variables:

Let $j$ = John's current age

Let $m$ = Mary's current age

2 Write equations:

$j = m + 4$ ...(John is 4 years older)

$(j + 6) + (m + 6) = 40$ ...(sum in 6 years)

Simplifying: $j + m = 28$ ...(2)

3 Substitute (1) into (2):

\((m + 4) + m = 28\]

\[2m + 4 = 28\]

\[2m = 24\]

\[m = 12\]

4 Find John's age:

\[j = 12 + 4 = 16\]

5 Answer: John is 16 years old and Mary is 12 years old.

Practice Problems

Question 1: Solve the simultaneous equations:

\[4x + 3y = 17\]

\[2x - y = 3\]

Show Solution

From equation (2): $y = 2x - 3$

Substitute into (1): $4x + 3(2x - 3) = 17$

$4x + 6x - 9 = 17$

$10x = 26$, so $x = 2.6$

$y = 2(2.6) - 3 = 2.2$

Solution: x = 2.6, y = 2.2

Question 2: The sum of two numbers is 15 and their difference is 3. Find the numbers.

Show Solution

Let the numbers be $x$ and $y$.

$x + y = 15$ ...(1)

$x - y = 3$ ...(2)

Adding: $2x = 18$, so $x = 9$

From (1): $y = 15 - 9 = 6$

The numbers are 9 and 6

Question 3: 3 books and 2 pens cost $21. 2 books and 5 pens cost $20. Find the cost of one book and one pen.

Show Solution

Let $b$ = cost of a book, $p$ = cost of a pen

$3b + 2p = 21$ ...(1)

$2b + 5p = 20$ ...(2)

Multiply (1) by 5: $15b + 10p = 105$ ...(3)

Multiply (2) by 2: $4b + 10p = 40$ ...(4)

Subtract (4) from (3): $11b = 65$, so $b = \frac{65}{11} \approx 5.91$

From (1): $p = \frac{21 - 3(5.91)}{2} \approx 1.64$

Book: $5.91, Pen: $1.64

Key Tips for CSEC Exams

Always label your equations (1), (2), etc.
Show all working clearly - marks are given for method
Check your answer by substituting back into BOTH original equations
For word problems, define your variables clearly first
Use elimination when coefficients are easy to match
Use substitution when one equation is already solved for a variable