3. using the principles together

OBJECTIVES

1.3 Using the Principles Together

dUse < or > for to

write a true statement in a situation like 6 10.

Slide 1Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

a Solve equations using both the addition principle andthe multiplication principle.

b Solve equations in which like terms may need to becollected.

c Solve equations by first removing parentheses andcollecting like terms; solve equations with an infinitenumber of solutions and equations with no solutions.

EXAMPLE

1.3 Using the Principles Together

aSolve equations using both the addition principle andthe multiplication principle.

Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.3 Using the Principles Together

a

Solve equations using both the addition principle andthe multiplication principle.

Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.3 Using the Principles Together

aSolve equations using both the addition principle andthe multiplication principle.

Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.3 Using the Principles Together

b

Solve equations in which like terms may need to becollected.

Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

If there are like terms on one side of the equation, we collect them before using the addition principle or the multiplication principle.

EXAMPLE

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 6Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.3 Using the Principles Together

b Solve equations in which like terms may need to becollected.

Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

If there are like terms on opposite sides of the equation, we get them on the same side by using the addition principle. Then we collect them. In other words, we get all the terms with a variable on one side of the equation and all the terms without a variable on the other side.

If there are like terms on one side at the outset, they should be collected first.

EXAMPLE

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 9Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 10Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

In general, equations are easier to solve if they do not contain fractions or decimals.

The easiest way to clear an equation of fractions is to multiply every term on both sides by the least common multiple of all the denominators.

EXAMPLE

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The denominators are 3, 6, and 2. The number 6 is the least common multiple of all the denominators. We multiply by 6 on both sides of the equation.

EXAMPLE

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.3 Using the Principles Together

b Solve equations in which like terms may need to becollected.

Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 14Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

To clear an equation of decimals, we count the greatest number of decimal places in any one number. If the greatest number of decimal places is 1, we multiply every term on both sides by 10; if it is 2, we multiply by 100; and so on.

EXAMPLE

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 16Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

The greatest number of decimal places in any one number is two. Multiplying by 100, which has two 0’s, will clear all decimals.

EXAMPLE

1.3 Using the Principles Together

bSolve equations in which like terms may need to becollected.

Slide 17Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.3 Using the Principles Together

c

Solve equations by first removing parentheses andcollecting like terms; solve equations with an infinitenumber of solutions and equations with no solutions.

Slide 18Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

To solve certain kinds of equations that contain parentheses, we first use the distributive laws to remove the parentheses. Then we proceed as before.

EXAMPLE

1.3 Using the Principles Together

cSolve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions.

Slide 19Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.3 Using the Principles Together

c

Solve equations by first removing parentheses and

collecting like terms; solve equations with an infinite

number of solutions and equations with no solutions.

Slide 20Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1. Multiply on both sides to clear the equation of fractions or decimals. (This is optional, but it can ease computations.)2. If parentheses occur, multiply to remove them using the distributive laws.3. Collect like terms on each side, if necessary.4. Get all terms with variables on one side and all numbers (constant terms) on the other side, using the addition principle.

1.3 Using the Principles Together

c

Solve equations by first removing parentheses and

collecting like terms; solve equations with an infinite

number of solutions and equations with no solutions.

Slide 21Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

5. Collect like terms again, if necessary.6. Multiply or divide to solve for the variable, using the multiplication principle.7. Check all possible solutions in the original equation.

EXAMPLE

1.3 Using the Principles Together

cSolve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions.

Slide 22Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.3 Using the Principles Together

cSolve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions.

Slide 23Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.3 Using the Principles Together

cSolve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions.

Slide 24Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.3 Using the Principles Together

cSolve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions.

Slide 25Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

1.3 Using the Principles Together

c

Solve equations by first removing parentheses and

collecting like terms; solve equations with an infinite

number of solutions and equations with no solutions.

Slide 26Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

EXAMPLE

1.3 Using the Principles Together

cSolve equations by first removing parentheses and collecting like terms; solve equations with an infinite number of solutions and equations with no solutions.

Slide 27Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.