Linear Equations

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Quick recall

• Use substitution or elimination, then check both original equations.
• A unique solution requires a nonzero determinant.

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Formula Cards

1. General linear form

$$ax+by=c$$

2. Substitution

Solve one equation for one variable, substitute into the other, then back-substitute.

3. Elimination

Match the coefficients of one variable, add or subtract to eliminate it, then back-substitute.

4. Unique solution condition

$$a_1b_2-a_2b_1\ne0$$

For a₁x+b₁y=c₁ and a₂x+b₂y=c₂. This determinant condition also works when some coefficients are zero.

Worked Examples

Example 1

Question: Solve 2x+y = 9 and x+y = 6.

Step 1: Subtract the second equation from the first.

$$(2x+y)-(x+y)=9-6\quad\Rightarrow\quad x=3$$

Step 2: Substitute into x+y = 6.

$$3+y=6\quad\Rightarrow\quad y=3$$

Answer: x = 3, y = 3.

Example 2

Question: Solve 3x+2y = 12 and x−y = 1.

Step 1: Rearrange the second equation.

$$x=y+1$$

Step 2: Substitute into the first.

$$3(y+1)+2y=12\quad\Rightarrow\quad5y=9$$

Step 3: Find both variables.

$$y=1.8,\qquad x=1.8+1=2.8$$

Answer: x = 2.8, y = 1.8.

Example 3

Question: Two numbers have sum 25 and difference 5. Find them.

Step 1: Translate into simultaneous equations.

$$x+y=25,\qquad x-y=5$$

Step 2: Add to eliminate y.

$$2x=30\quad\Rightarrow\quad x=15$$

Step 3: Find y.

$$y=25-15=10$$

Answer: 15 and 10.

Drill

Try each question before opening its answer.

Question 1

Solve: $x+y=10$, $x-y=2$.

• Show answer

  x=6, y=4

Question 2

Solve: $2x+3y=13$, $x+y=5$.

• Show answer

  x=2, y=3

Question 3

The sum of two numbers is 40 and one is 3 times the other. Find them.

• Show answer

  10, 30

Question 4

Solve: $4x-y=7$, $2x+y=11$.

• Show answer

  x=3, y=5

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