1.2 Solving Two-Step Linear Equations

SECTION 1.2 Learning Objective

1.2: Solving Two-Step Linear Equations

  • Use properties of equality to isolate variables and solve algebraic equations

 

In the previous section, we used the addition and multiplication properties of equality to solve algebraic equations.  In this section, we will look at equations where both properties are needed in order to solve the equation.

Addition Property of Equality

For all real numbers a, b, and c: If [latex]a=b[/latex], then [latex]a+c=b+c[/latex].

If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.

Multiplication Property of Equality

For all real numbers a, b, and c: If a = b, then [latex]a\cdot{c}=b\cdot{c}[/latex] (or ab = ac).

If two expressions are equal to each other and you multiply both sides by the same number, the resulting expressions will also be equivalent.

Use properties of equality to isolate variables and solve algebraic equations

steps leading to a gold ball

Steps With an End In Sight

There are some equations that you can solve in your head quickly. For example—what is the value of y in the equation [latex]2y=6[/latex]? Chances are you didn’t need to get out a pencil and paper to calculate that [latex]y=3[/latex]. You only needed to do one thing to get the answer: divide 6 by 2.

Other equations are more complicated. Solving [latex]\displaystyle 4\left( \frac{1}{3}t+\frac{1}{2}\right)=6[/latex] without writing anything down is difficult! That’s because this equation contains not just a variable but also fractions and terms inside parentheses. This is a multi-step equation, one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.

Remember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The addition property of equality and the multiplication property of equality explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you’ll keep both sides of the equation equal.

If the equation is in the form [latex]ax+b=c[/latex], where x is the variable, you can solve the equation as before. First “undo” the addition and subtraction, and then “undo” the multiplication and division.

Example 1

Solve [latex]3y+2=11[/latex].

In this next video you will see a few more examples of solving two-step linear equations.