Lesson 4: 2-4 Solving equations with variables on each side

What operations are needed to isolate the variable terms on one side of the equation?

When does an equation have no solution?

When is an equation true for all values of the variable?

What is the first step to solving an equation with the variable on each side?

identity

A.CED.1 Create equations and inequalities in one variable and use them to solve problems.

A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

3A Functions
3B Functions
3C Functions

To solve equations with variables on each side, first use the Distributive Property (if applicable) and then simplify using the properties of equality. If all of the variable terms are eliminated as a result, and both sides of the equation are different, then the equation is not true, and there is no solution. If both sides of the equation are identical, the equation is an identity, and all values are solutions of this equation.

The equation y = 13x + 19 represents the number of times Americans eat in their cars each year, where x is the number of years since 1985, and y is the number of times that they eat in their car. The equation y = -13x + 93 represents the number of times Americans eat in restaurants each year, where x is the number of years since 1985 and y is the number of times that they eat in a restaurant.

The equation 13x + 19 = -13x +93 represents the year when the number of times American eat in their cars will equal the number of times Americans eat in restaurants.

Eliminating the Variable: Mathematically proficient students detect possible errors by strategically using estimation and other mathematical knowledge. Emphasize that there are only two possible outcomes under which the variable can be eliminated from an equation: either the equation has no solution (false statement) or the equation is an identity (true statement).

If some students are having trouble solving equations with a variable on each side, then those students may benefit from using an equation mat and algebra tiles. Have students model the equation, and then get them started by asking what they must do to remove the x-tiles from each side of the equation mat. Use questions to focus their attention on isolating the variable. Have them write out the steps they used after they solve for x.

Extension: Remind students that they have learned that an equation can have 1, 0, or an endless number of solutions. Ask students to write an equation in which there are exactly two solutions for the variable. For example, in |x| = 25, x = 25 or –25 or in x2 = 25, x = 5 or –5.

Exercises 48 - 56

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