1.6 Multi-Step Equations

Leanne Thompson

1.6 Multi-Step Equations

In the examples in the previous lesson, we were able to isolate the variable with just one or two operations. Many of the equations that we are asked to solve will take more than two steps to solve. When there are several steps that need to be completed to solve an equation, we will usually need to simplify one or both sides of the equation before starting to isolate the variable.

The first multi-step equations that we solve in this lesson will require us to put terms with variables on one side of the equation and terms that are constants on the other side of the equation.

Example 1

Solve: $7x+5=6x+2$

Table 1.6.1
Steps	Solution
Collect the variable terms on one side of the equation by subtracting 6x from both sides.	$7x+5=6x+2$ $7x-6x+5=6x-6x+2$
Simplify.	$x+5=2$
Collect the constants on the other side of the equation by subtracting 5 from both sides.	$x+5-5=2-5$
Simplify.	$x=-3$
Verify the solution.	$(7)(-3)+5=(6)(-3)+2$ $-21+5=-18+2$ $-16=-16$ x = –3 is the solution.

Practice 1

Solve.

a) $12x+8=6x+2$

b) $6n-2=-3n+7$

c) $2a-7=5a+8$

The following steps can be followed to solve more complicated equations. Depending on the question, you may not need to complete all the steps.

Table 1.6.2
To solve a multi-step equation:
Step 1: Use the Distributive Property to remove any parentheses. Step 2: Simplify each side of the equation as much as possible by combining like terms. Step 3: Collect all the variable terms on one side of the equation. Step 4: Collect all the constant terms on the other side of the equation. Step 5: Make the coefficient of the variable term equal to 1 to fully isolate for the unknown value. Step 6: Verify the solution.

Example 2

Solve: $3\left(x+2\right)=18$ .

Table 1.6.3
Steps	Solution
Use the Distributive Property to remove any parentheses.	$3\left(x+2\right)=18$ $3x+6=18$
Collect all the constant terms on the side of the equation that the term with the variable is not on. Simplify.	$3x+6-6=18-6$ $3x=12$
Make the coefficient of the variable term equal to 1 to fully isolate the unknown value. Simplify.	$3x\div3=12\div3$ $x=4$
Verify the solution.	$3\left(4+2\right)=18$ $3\left(6\right)=18$ $18=18$ x = 4 is the solution.

Practice 2

Solve.

a) $6\left(y-4\right)=-18$

b) $-\left(x+5\right)=7$

c) $8x+9x-5x=-3+15$

d) $4\left(3x-2\right)=5\left(2x+4\right)$

e) $5\left(x-1\right)=8\left(1-x\right)$

f) $8-2\left(3y+5\right)=0$

Homework

Solve.

a) $9y+4=7y+12$

b) $8q-5=-4q+7$

c) $7n-3=n+3$

d) $2a-2=6a+18$

e) $4k-1=7k+17$

f) $3.4x+4=1.6x-5$

g) $2.8x+12=-1.4x-9$

h) $3.6y+8=1.2y-4$

i) $5\left(x+3\right)=35$

j) $-\left(y+8\right)=-2$

k) $-\left(z+4\right)=-12$

l) $2\left(a-4\right)+3=-1$

m) $7\left(n-3\right)-8=-15$

n) $12-3\left(4j+3\right)=-17$

o) $-6-8\left(k-2\right)=-10$

p) $7x+6x-4x=-8+26$

q) $11-20=17y-8y-6y$

r) $18-22=12x-x-4x$

s) $-3\left(n-2\right)-6=21$

t) $3\left(n+2\right)=2\left(n+5\right)$

u) $3\left(2p+4\right)=4\left(p-1\right)$

v) $2\left(3d-7\right)=-3\left(-4d+2\right)$

w) $7\left(3m-2\right)=-4\left(6-5m\right)$

x) $3\left(n-4\right)-2n=-3$
Solve for x: $5\left(2x-3\right)=3\left(4x+1\right)+7$

a) x = $-\frac{1}{2}$

b) x = $\frac{9}{2}$

c) x = $\frac{3}{2}$

d) x = $-\frac{25}{2}$
Solve for x: $4x-7=2\left(3x+5\right)-3$

a) x = –7
b) x = 3
c) x = –15
d) x = 11

Answers

1.

a) 4	b) 1	c) 1	d) –5	e) –6	f) –5
g) –5	h) –5	i) 4	j) –6	k) 8	l) 2
m) 2	n) $\frac{5}{3}$	o) $\frac{5}{2}$	p) 2	q) –3	r) $-\frac{4}{7}$
s) –7	t) 4	u) –8	v) $-\frac{4}{3}$	w) –10	x) 9