Lesson 15: Solving Equations with Rational Numbers

Solving Equations with Rational Numbers

Student Summary

To solve the equation $x + 8 = \text-5$ , we can add the opposite of 8, or -8, to each side:

Because adding the opposite of a number is the same as subtracting that number, we can also think of it as subtracting 8 from each side.

$\begin{aligned} x + 8 &= \text-5\\ (x+ 8) + \text-8&=(\text-5)+ \text-8\\ x&=\text-13 \end{aligned}$

We can use the same approach for this equation:

$\begin{aligned} \text-12 & = t +\text- \frac29\\ (\text-12)+ \frac29&=\left( t+\text-\frac29\right) + \frac29\\\text-11\frac79& = t\end{aligned}$

To solve the equation $8x = \text-5$ , we can multiply each side by the reciprocal of 8, or $\frac18$ :

Because multiplying by the reciprocal of a number is the same as dividing by that number, we can also think of it as dividing by 8.

$\begin{aligned} 8x & = \text-5\\ \frac18 ( 8x )&= \frac18 (\text-5)\\ x&=\text-\frac58 \end{aligned}$

We can use the same approach for this equation:

$\begin{aligned} \text-12& =\text-\frac29 t\\ \text-\frac92\left( \text-12\right)&= \text-\frac92 \left(\text-\frac29t\right) \\ 54& = t\end{aligned}$

Visual / Anchor Chart

Standards

Building On

7.NS.A

No additional information available.

6.EE.5

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

Addressing

7.NS.3

Solve real-world and mathematical problems involving the four operations with rational numbers.

7.EE.4

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

7.EE.3

Solve multi-step real-world and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Assess the reasonableness of answers using mental computation and estimation strategies.

7.NS.A

No additional information available.

Building Toward

7.EE.B

No additional information available.

7.EE.4.a

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.