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1. Equations & Inequalities in One Variable
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United States of AmericaTennessee
Integrated Math 1

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+3xDistributive property
\displaystyle -24\displaystyle =\displaystyle -24Subtraction property of equality

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+5Multiplication property of equality
\displaystyle 9\displaystyle =\displaystyle 5Subtraction property of equality

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+85Distributive property
\displaystyle 35+3x\displaystyle =\displaystyle 85Subtraction property of equality

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+6Distributive property
\displaystyle 3x-36\displaystyle =\displaystyle 6Subtraction property of equality
\displaystyle 3x\displaystyle =\displaystyle 42Addition property of equality
\displaystyle x\displaystyle =\displaystyle 14Division property of equality

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

Solve the following equation: \dfrac{(x+3)}{2}+1.3x=\dfrac{0.9x}{4}

Solution

\displaystyle \dfrac{(x+3)}{2}+1.3x\displaystyle =\displaystyle \dfrac{0.9x}{4}
\displaystyle 2\left(x+3\right)+5.2x\displaystyle =\displaystyle 0.9xMultiplication property of equality
\displaystyle 2x+6+5.2x\displaystyle =\displaystyle 0.9xDistributive property
\displaystyle 7.2x+6\displaystyle =\displaystyle 0.9xCombine like terms
\displaystyle 6\displaystyle =\displaystyle -6.3xSubtraction property of equality
\displaystyle -1.05\displaystyle =\displaystyle xDivision property of equality

Example 4

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-10Distributive property
\displaystyle y+56\displaystyle =\displaystyle 5yAddition property of equality
\displaystyle 56\displaystyle =\displaystyle 4ySubtraction property of equality
\displaystyle 14\displaystyle =\displaystyle yDivision property of equality

Bianca is currently 14 years old.

Outcomes

M1.N.Q.A.1

Use units as a way to understand real-world problems.*

M1.N.Q.A.1.C

Define and justify appropriate quantities within a context for the purpose of modeling.*

M1.A.CED.A.1

Create equations and inequalities in one variable and use them to solve problems in a real-world context.*

M1.A.REI.A.1

Understand solving equations as a process of reasoning and explain the reasoning. Construct a viable argument to justify a solution method.

M1.A.REI.B.2

Solve linear and absolute value equations and inequalities in one variable.

M1.A.REI.B.2.A

Solve linear equations and inequalities, including compound inequalities, in one variable. Represent solutions algebraically and graphically.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

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