Topics
1. Equations & Inequalities in One Variable
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United States of AmericaFL
Grade 912

1.03 Equations with variables on both sides

Lesson
Practice
Lesson

Concept summary

The first step in solving equations with variables on both sides is usually to move all variable terms to one side of the equation, by applying the properties of equality to variable terms.

A fully simplified equation in one variable will take one of the following three forms, corresponding to how many solutions the equation has:

  • x=a, where a is a number (a unique solution)

  • a=a, where a is a number (infinitely many solutions)

  • a=b, where a and b are different numbers (no solutions)

An equation of the second form, which is true for any possible value of the variable(s), is sometimes called an identity.

Worked examples

Example 1

Determine how many solutions the following equations have:

a

3(-8+x)=3(-8+x)

Solution

\displaystyle 3(-8+x)\displaystyle =\displaystyle 3(-8+x)
\displaystyle -24+3x\displaystyle =\displaystyle -24+3xDistribute the 3
\displaystyle -24\displaystyle =\displaystyle -24Subtract 3x from both sides

Since the equation is of the form a=a, where a is a number, there are infinitely many solutions.

b

\dfrac{9+x}{9}=\dfrac{x+5}{9}

Solution

\displaystyle \dfrac{9+x}{9}\displaystyle =\displaystyle \dfrac{x+5}{9}
\displaystyle 9+x\displaystyle =\displaystyle x+5Multiply both sides by 9
\displaystyle 9\displaystyle =\displaystyle 5Subtract x from both sides

Since the equation is of the form a=b, where a and b are different numbers, there are no solutions.

c

5(7+x)=2x+85

Solution

\displaystyle 5(7+x)\displaystyle =\displaystyle 2x+85
\displaystyle 35+5x\displaystyle =\displaystyle 2x+85Distribute the 5
\displaystyle 35+3x\displaystyle =\displaystyle 85Subtract 2x from both sides

This equation will simplify to one of the form x=a, where a is a number, so there is a unique solution.

Reflection

We did not have to solve the equation to determine that it has a unique solution - we only needed to check that the variable terms did not cancel each other out, like they did in part (a) and part (b).

Example 2

Solve the following equation: 4(x-9)=x+6

Solution

\displaystyle 4(x-9)\displaystyle =\displaystyle x+6
\displaystyle 4x-36\displaystyle =\displaystyle x+6Distribute the 4
\displaystyle 3x-36\displaystyle =\displaystyle 6Subtract x from both sides
\displaystyle 3x\displaystyle =\displaystyle 42Add 36 to both sides
\displaystyle x\displaystyle =\displaystyle 14Divide both sides by 3

Reflection

We can check if x=14 is correct by substituting 14 in for x into each side and checking that they are equal:

\text{LHS}=4\left(14-9\right)=4\cdot5=20

\text{RHS}=14+6=20

Example 3

Right now, Bianca's father is 48 years older than Bianca.

2 years ago, her father was 5 times older than her.

Solve for y, Bianca's current age.

Approach

We want to write expressions that represent Bianca's age and her father's age. Then we relate them with an equation and solve for y.

Bianca's father is currently y+48 years old.

Two years ago, Bianca was y-2 years old. Her father was y+48-2=y+46 years old.

Her father's age was five times her age at this time, which produces our equation:

y+46=5(y-2)

Solution

\displaystyle y+46\displaystyle =\displaystyle 5(y-2)
\displaystyle y+46\displaystyle =\displaystyle 5y-10Distribute the 5
\displaystyle y+56\displaystyle =\displaystyle 5yAdd 10 to both sides
\displaystyle 56\displaystyle =\displaystyle 4ySubtract y from both sides
\displaystyle 14\displaystyle =\displaystyle yDivide both sides by 4

Bianca is currently 14 years old.

Outcomes

MA.912.AR.1.1

Identify and interpret parts of an equation or expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.

MA.912.AR.2.1

Given a real-world context, write and solve one-variable multi-step linear equations.

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