Algebra I (MAT111): Two-Step Equations with Addition and Division (practice)

Guidance

Let's take a look at how to solve a two-step equation.

When we perform inverse operations to find the value of a variable, we work to get the variable alone on one side of the equals. This is called isolating the variable. It is one strategy for solving equations. You can use isolating the variable whether you are solving one-step or two-step equations.

Here is a two-step equation.

Solve for $c$ : $5 + \frac{c}{4} = 15$ .

We can call each piece of the equation a term . There is a term with a variable and there is a term without a variable. Notice that there are two terms on the left side of the equation, 5 and $\frac{c}{4}$ .

First Step of the Term

Our first step should be to use inverse operations to get the term that includes a variable, $\frac{c}{4}$ , by itself on one side of the equal (=) sign.

In the equation, 5 is added to $\frac{c}{4}$ . So, we can use the inverse of addition—subtraction. We can subtract 5 from both sides of the equation.

$5 + \frac{c}{4} & = 15\\5 - 5 + \frac{c}{4} & = 15 - 5\\0 + \frac{c}{4} & = 10\\\frac{c}{4} & = 10$

Second Step of the Term

Now, the term that includes a variable, $\frac{c}{4}$ , is by itself on one side of the equation.

We can now use inverse operations to get the $c$ by itself. Since $\frac{c}{4}$ means $c \div 4$ , we can use the inverse of division—multiplication. We can multiply both sides of the equation by 4.

$\frac{c}{4} & = 10\\\frac{c}{4} \times 4 & = 10 \times 4\\\frac{c}{4} \times \frac{4}{1} & = 40$

The number 4, or $\frac{4}{1}$ , is the multiplicative inverse , or reciprocal , of $\frac{1}{4}$ . You can find the multiplicative inverse of a number by flipping its numerator and its denominator. So, the multiplicative inverse of $\frac{4}{1}$ is $\frac{1}{4}$ . When a number is multiplied by its multiplicative inverse, the product is 1.

$\frac{c}{4} \times \frac{4}{1} & = 40\\c \times \left ( \frac{1}{4} \times \frac{4}{1} \right ) & = 40\\c \times 1 & = 40\\c & = 40$

The work above shows how multiplying each side of the equation by 4 isolates the variable.

Because 4 is the multiplicative inverse, or reciprocal, of $\frac{1}{4}$ , we could also have solved this problem by canceling out the 4's like this:

$\frac{c}{\bcancel{4}} \times \frac{\bcancel{4}}{1} & = 40\\\frac{c}{1} & = 40\\c & = 40$

The answer is that $c$ is equal to 40.

Instructions:

Work the problems out on paper before entering answers into Moodle. Bring your work to school because we will discuss your work. This Moodle practice quiz will provide feedback on how to solve the problems. So you are allows to retake as many times as necessary for you understand the two-step equations with addition and division. (note the questions will change)

Grading method: Highest grade

Guidance

Bismarck Standards this activity aligns:

Here is a two-step equation.

First Step of the Term

Second Step of the Term

Instructions: