Solving Multi-Step Equations

Learning Objectives

  • Solve multi-step equations

We solve equations using properties of real numbers and properties of equality.

Properties of Equality

For two expressions [latex]S[/latex] and [latex]T[/latex] and any constant [latex]c[/latex]:

Addition Property of Equality: If [latex] S=T[/latex] then [latex]S+c=T+c[/latex]

Multiplication Property of Equality: If [latex]S=T[/latex] then [latex]S \cdot c = T \cdot c[/latex], provided [latex]c \neq 0[/latex]

These properties tell us we can add an expression to both sides of an equation and multiply each side of an equation by a nonzero expression to obtain an equivalent equation.

Example

Solve each equation:

  1. [latex]x+1=7[/latex]
  2. [latex]3x=45[/latex]

Solving Multi-step Equations

  1. (Optional) Multiply to clear any fractions or decimals.
  2. Simplify each side by clearing parentheses and combining like terms.
  3. Add or subtract to isolate the variable term—possibly a term with the variable.
  4. Multiply or divide to isolate the variable.
  5. Check the solution.

Example

Solve for [latex]a[/latex].

[latex]4\left(2a+3\right)=28[/latex]

In the video that follows, we show another example of how to use the distributive property to solve a multi-step linear equation.

In the next example we have parenthesis on each side of the equation, so we need to apply the distributive property on each side.

Example

Solve for [latex]t[/latex].

[latex]2\left(4t-5\right)=-3\left(2t+1\right)[/latex]

In the following video, we solve another multi-step equation with two sets of parentheses.

Try It

If an equation contains fractions, it is sometimes helpful to clear fractions by multiplying both sides of the equation by the lowest common denominator of all fractions in the equation.

Example

Solve  [latex]\frac{1}{2}x-3=2-\frac{3}{4}x[/latex] by clearing the fractions in the equation first.

In the following video, we show how to solve a multi-step equation with fractions.