Lesson 3 Solving Equations Literally Practice Understanding

Learning Focus

Compare strategies for solving linear equations and literal equations.

How is solving an equation with one variable similar to the work of solving an equation with more than one variable?

Open Up the Math: Launch, Explore, Discuss

Notice and Wonder

Solve each of the following equations for $x :$

1.

$\frac{3 x + 2}{5} = 7$

2.

$\frac{3 x + 2 y}{5} = 7$

3.

$\frac{4 x}{3} - 5 = 11$

4.

$\frac{4 x}{3} - 5 y = 11$

5.

$\frac{2}{5} (x + 3) = 6$

6.

$\frac{2}{5} (x + y) = 6$

7.

$2 (3 x + 4) = 4 x + 12$

8.

$2 (3 x + 4 y) = 4 x + 12 y$

Write a verbal description for each step of the equation-solving process used to solve the following equations for $x$ . Your description should include statements about how you know what to do next. For example, you might write, “First I . The reason I can do this is because of .”

9.

$\frac{a x + b}{c} - d = e$

10.

$r \cdot \sqrt{\frac{m x}{n} + s} = t$

Ready for More?

Work with a partner to revise and refine your justifications for problems 9 and 10. For example, what properties might you use to convince yourself that $\frac{1}{r} \cdot r \cdot \sqrt{\frac{m x}{n} + s} = \frac{1}{r} \cdot t$ is equivalent to $\sqrt{\frac{m x}{n} + s} = \frac{t}{r}$ ?

Takeaways

Today I gained some new insights into the process for solving equations, including:

When solving literal equations, we may need to change the form of the expressions involving variables and operations. The following properties of operations justify how we might rewrite these expressions:

Properties of Addition
Additive Identity	Additive Inverse	Commutative	Associative
Properties of Multiplication
Multiplicative Identity	Multiplicative Inverse	Commutative	Associative
The Distributive Property

Vocabulary

Lesson Summary

In this lesson, we compared strategies for solving linear equations and literal equations and found that the processes for solving each are similar. We solve both types of equations by using inverse operations in the reverse order from the order used when evaluating the expression that involves the variable we are solving for. However, the answer to a linear equation is a number, while the answer to a literal equation is a variable or an expression. Sometimes we have to combine like terms, particularly if expressions containing the same variable occur on both sides of the equations. Properties of operations and properties of equality guide our thinking when solving equations and help us justify each step in our equation-solving process.

Retrieval

Perform the indicated operation on both sides of the given inequality and then decide if the new inequality you create is true or false.

1.

Given: $7 > - 3$ . Add $5$ to both sides.

2.

Given: $- 8 < 12$ . Multiply both sides by $2$ .

3.

Given: $6 < 10$ . Divide both sides by $- 2$ .

Identify the $x$ -intercepts and the $y$ -intercepts in the representations and write them as coordinate pairs.

4.

$x$	$y$
$- 4$	$12$
$- 3$	$8$
$- 2$	$4$
$- 1$	$0$
$0$	$- 4$
$1$	$- 8$
$2$	$- 12$

$x$ -intercept:

$y$ -intercept:

5.

$x$ -intercept:

$y$ -intercept:

6.

Solve the equation and justify each step.

$3 x + 6 = x + 14$