4.1 Solving Equations

Whatever you do to one side of the equation, you must do to the other side.

Step 1: Subtract 5 from both sides
$$3x + 5 - 5 = 14 - 5$$ $$3x = 9$$
Step 2: Divide both sides by 3
$$x = \frac{9}{3} = 3$$
Check: $3(3) + 5 = 9 + 5 = 14$ ✓

Step 1: Get all $x$ terms on one side (subtract $2x$)
$$7x - 2x - 4 = 11$$ $$5x - 4 = 11$$
Step 2: Add 4 to both sides
$$5x = 15$$
Step 3: Divide by 5
$$x = 3$$

Step 1: Subtract $3x$ from both sides
$$-3 = 6x + 7$$
Step 2: Subtract 7 from both sides
$$-10 = 6x$$
Step 3: Divide by 6
$$x = \frac{-10}{6} = -\frac{5}{3}$$

Step 1: Expand the bracket
$$3x + 6 = 15$$
Step 2: Subtract 6 from both sides
$$3x = 9$$
Step 3: Divide by 3
$$x = 3$$

Step 1: Expand both brackets
$$6x - 2 = 4x + 12$$
Step 2: Subtract $4x$ from both sides
$$2x - 2 = 12$$
Step 3: Add 2 to both sides
$$2x = 14$$
Step 4: Divide by 2
$$x = 7$$

Step 1: Expand both brackets (careful with the negative!)
$$10x + 5 - 3x + 6 = 20$$
Step 2: Collect like terms
$$7x + 11 = 20$$
Step 3: Subtract 11
$$7x = 9$$
Step 4: Divide by 7
$$x = \frac{9}{7}$$

Step 1: Multiply both sides by 4
$$x = 20$$

Step 1: Multiply both sides by 3
$$2x + 1 = 15$$
Step 2: Subtract 1
$$2x = 14$$
Step 3: Divide by 2
$$x = 7$$

Step 1: Find the LCM of 2 and 3 = 6
Step 2: Multiply every term by 6
$$\frac{6x}{2} + \frac{6x}{3} = 60$$ $$3x + 2x = 60$$
Step 3: Simplify and solve
$$5x = 60$$ $$x = 12$$

Step 1: Cross multiply
$$3(x + 2) = 4(x - 1)$$
Step 2: Expand
$$3x + 6 = 4x - 4$$
Step 3: Rearrange
$$6 + 4 = 4x - 3x$$ $$10 = x$$

Step 1: Make the coefficients of one variable the same
Step 2: Add or subtract the equations to eliminate that variable
Step 3: Solve for the remaining variable
Step 4: Substitute back to find the other variable

Step 1: Add the equations (y coefficients are +1 and -1)
$(2x + y) + (x - y) = 7 + 2$
$3x = 9$
$x = 3$

Step 2: Substitute $x = 3$ into equation (1)
$2(3) + y = 7$
$6 + y = 7$
$y = 1$

Answer: $x = 3, y = 1$

Step 1: Subtract equation (2) from (1) (both have $2y$)
$(3x + 2y) - (x + 2y) = 12 - 8$
$2x = 4$
$x = 2$

Step 2: Substitute into equation (2)
$2 + 2y = 8$
$2y = 6$
$y = 3$

Answer: $x = 2, y = 3$

Step 1: Make y coefficients the same
Multiply (1) by 2: $4x + 6y = 26$
Multiply (2) by 3: $15x - 6y = 12$

Step 2: Add the equations
$19x = 38$
$x = 2$

Step 3: Substitute into (1)
$2(2) + 3y = 13$
$4 + 3y = 13$
$3y = 9$
$y = 3$

Answer: $x = 2, y = 3$

Step 1: Rearrange one equation to make one variable the subject
Step 2: Substitute this expression into the other equation
Step 3: Solve the resulting equation
Step 4: Substitute back to find the other variable

Step 1: Equation (1) already has $y$ as the subject

Step 2: Substitute $y = 2x + 1$ into equation (2)
$3x + (2x + 1) = 11$
$5x + 1 = 11$
$5x = 10$
$x = 2$

Step 3: Substitute $x = 2$ into equation (1)
$y = 2(2) + 1 = 5$

Answer: $x = 2, y = 5$

Step 1: Rearrange (1) to make $x$ the subject
$x = 5 - y$

Step 2: Substitute into equation (2)
$2(5 - y) - 3y = -5$
$10 - 2y - 3y = -5$
$10 - 5y = -5$
$-5y = -15$
$y = 3$

Step 3: Find $x$
$x = 5 - 3 = 2$

Answer: $x = 2, y = 3$

Use Elimination when:
• Coefficients are already similar or easy to match
• Neither equation has an obvious rearrangement

Use Substitution when:
• One equation already has a variable as the subject
• One variable has a coefficient of 1 (easy to rearrange)

$<$ means "less than"

$>$ means "greater than"

$\leq$ means "less than or equal to"

$\geq$ means "greater than or equal to"

When you multiply or divide by a negative number, you must REVERSE the inequality sign.

$-2x > 6$
$x < -3$ (sign flipped when dividing by $-2$)

Step 1: Subtract 5 from both sides
$3x > 9$

Step 2: Divide by 3
$x > 3$

Solution: All values greater than 3

Step 1: Subtract 4 from both sides
$-2x \geq 6$

Step 2: Divide by $-2$ (FLIP THE SIGN!)
$x \leq -3$

Solution: All values less than or equal to $-3$

Step 1: Subtract 1 from all three parts
$-6 < 2x \leq 6$

Step 2: Divide all parts by 2
$-3 < x \leq 3$

Solution: $x$ is greater than $-3$ AND less than or equal to 3

• Open circle ○ means the value is NOT included ($<$ or $>$)

• Closed circle ● means the value IS included ($\leq$ or $\geq$)

• Draw an arrow in the direction of the solution