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# Solving Linear Equations

Consider the equation:

3x − 5 =13

We are undoing what is happening to x ( Multiplication by 3 and subtraction
by 5 ) . So we undo by going backwards: undo subtraction of 5 and then
undo multiplication by 3. Any equation that can be written in the form ax + b = c is called a linear
equation. In this section all of the equations are linear. To solve linear
equations:

1.) Simplify both sides of the equation.
2.) Collect all the variable terms and all constant terms on the

other side.

3.) Isolate the variable and solve.

EXAMPLE: Solve and Check To Solve we can do two things: multiply by 3 or multiply by 2x = ____ divide by 2 x = ____
x = ___

EXAMPLE: Solve and Check Sometimes it may be necessary to simplify before trying to solve. We get all
the x 's together and the constants together before we solve.

EXAMPLE: Solve and Check Many people would love to do away with having to deal with fractions. When
an equation has fractions involved, we may use the multiplication property of
equality to clear the fractions. We multiply both sides by the LCD. Therefore we don’t need to
deal with fractions very much.

EXAMPLE: Solve: IDENTITIES AND IMPOSSIBLE EQUATIONS:

The equations that we have been solving are called conditional equations
because they only work for one number. An equation that is true for all values
of its variable is called an identity.

3( x + 2) = 3x + 6 is an identity

2x + 2 = 2x + 3 is an equation that is never true.

Solve the above equation to see what happens: EXAMPLE: Solve the equation.

 a.) −3( s + 2) = −2( s + 4) − s EXAMPLE: A driver left the plant with 300 bottles of drinking water on his truck. His route consisted of office buildings, each of which received 3 bottles of water. The driver returned to the plant with 117 bottles on the truck. To how many office buildings did he deliver? EXAMPLE: The formula: models the recommended weight, W , in pounds, for a male, where H represents the man’s height , in inches, over 5 feet. What is the recommended weight for a man who is 6 feet, 3 inches tall?