3 4 Solving Equations with Variables on Both

3. 4. Solving Equations with Variables on Both Sides

Collecting Variables on One Side Some equations have variables on both sides. To solve such equations, you may first collect the variable terms on the side with the greater variable coefficient.

Collect Variables on Left Side Solve Step One: Write the original equation.

Collect Variables on Left Side Solve Step Two: Look at the coefficients of the x-terms. Since 7 is greater than -2, collect the x-terms on the left side. Add 2 x to each side.

Collect Variables on Left Side Solve Step Three: Simplify

Collect Variables on Left Side Solve Step Four: Subtract 19 from each side.

Collect Variables on Left Side Solve Step Five: Simplify

Collect Variables on Left Side Solve Step Six: Divide each side by 9.

Collect Variables on Left Side Solve Step Six: Simplify

Collect Variables on Left Side Step Seven: Write original equation. Substitute 4 for each x. Solution is correct.

Collect Variables on Right Side Solve Step One: Write the original equation.

Collect Variables on Right Side Solve Step Two: Add 9 y to each side

Collect Variables on Right Side Solve Step Three: Simplify

Collect Variables on Right Side Solve Step Four: Divide each side by 15.

Collect Variables on Right Side Solve Step Five: Simplify.

What is an Identity? Linear equations do not always have one solution. An identity is an equation that is true for all values of the variable. Some linear equations have no solution. Lets look at a few examples…

Many Solutions Solve: Step One: Write the original equation.

Many Solutions Solve: Step Two: Use distributive property.

Many Solutions Solve: Step Three: Subtract 3 x from each side. All values of x are solutions, because 6 = 6 is always true. The original equation is an identity.