1.5 One-Step and Two-Step Equations

An equation is a mathematical statement of two equal expressions. In this section, we will use algebra to help us solve for the unknown value in an equation. When we solve equations, the main goal is to isolate the variable. As we isolate the variable, it is important to remember that whatever change is made to one side of the equation also must be made to the other side of the equation. This ensures that the equation stays equal.

To understand why you must change each side of the equation in the same way, you can think of an equation as a balance scale. To keep the scale balanced, or equal, you must add or take away the same amount of weight from both sides. Similarly, if you apply the same changes to both sides of an equation, it will stay balanced or equal. The images below illustrate this:

One weight on each side, so the scale is balanced.	One weight has been added to each side of the scale, so the scale remains balanced.	One weight has been added to the right side of the scale, so the scale becomes unbalanced.

Solving One-Step Equations

The following equations can be solved in one step. You will add, subtract, multiply, or divide the same value from each side of the equation to solve for x.

Table 1.5.1

Steps	Solution
Subtract 11 from each side. This will remove the +11 from the left side of the equation, and isolate x. Subtracting 11 from both sides ensures that the equation remains equal.
Simplify.
Verify the solution.	x = –14 is the solution.

Practice 1

Add or subtract a value from each side of the equation to solve for the unknown variable.

a)

b)

c)

d)

e)

f)

Table 1.5.2

Steps	Solution
Divide each side by 4. This will remove the 4 from the left side of the equation, and isolate x. Dividing both sides by 4 ensures that the equation remains equal.
Simplify.
Verify the solution.	x = –7 is the solution.

Practice 2

Multiply or divide each side of the equation by a value to solve for the unknown variable.

a)

b)

c)

d)

e)

f)

Table 1.5.3

Steps	Solution
Subtract 6 from each side. This will remove the 6 from the left side of the equation.
Simplify.
Divide each side by 4. This will remove the 4 from the left side of the equation and isolate x.
Simplify.
Verify the solution.	x = –5 is the solution.

Practice 3

Solve for the unknown variable in two steps.

a)

b)

c)

d)

e)

f)

Homework 

1. Solve.

   a)

   b)

   c)

   d)

   e)

   f)

   g)

   h)

   i)

   j)

   k)

   l)

   m)

   n)

   o)

2. Solve.

   a)

   b)

   c)

   d)

   e)

   f)

   g)

   h)

   i)

   j)

   k)

   l)

   m)

   n)

   o)

   p)

   q)

   r)

3. Solve.

   a)

   b)

   c)

   d)

   e)

   f)

   g)

   h)

   i)

   j)

   k)

   l)

   m)

   n)

   o)

   p)

   q)

   r)

4. Solve for x in the equation:

   a) x = 3 b) x = 5 c) x = 7 d) x = 15

5. You are solving the equation . What should you do first to isolate x?

   a) Add 5 to both sides of the equation. b) Subtract 5 from both sides of the equation. c) Multiply both sides by 4. d) Divide both sides by 4.

6. You are solving the equation . What two steps must you take to solve for x?

   a) First, add 6 to both sides, then divide both sides by –7. b) First, subtract 6 from both sides, then divide both sides by –7. c) First, add 6 to both sides, then multiply both sides by –7. d) First, subtract 7 from both sides, then divide both sides by –6.

Answers

1.

a) –16	b) –20	c) –4	d) –1	e)
f)	g) 6.4	h) 14	i) 5	j)
k) –11.7	l) 10	m)	n) 13.8	o)

2.

a) –16	b) –13	c) 128	d) –8	e) –3	f) 35
g) 18	h) 3	i) –6	j) 15	k) 0	l) 28
m) 36	n) –48	o) 80	p) 25	q) –32	r)

3.

a) –8	b) 5	c) 9	d) 10	e) –4	f) –7
g) –3	h) 3	i) 2	j) 1	k) –8	l) –8
m) –4	n) –2	o) –3	p) 3	q)	r) –2