Transforming Equations: Addition and Subtraction

September 21, 2011

Transforming Equations

Objective To solve equations using addition and subtraction.

Transforming Equations

Two soccer teams are tied at half time: 2 to 2. If each team scores 3 goals in the second half, the score will still be tied:

2 + 3 = 2 + 3

Two sporting goods stores charge $36 for a soccer ball. If, during a spring sale, each store reduces the price by $5, both stores will still be charging the same price:

36 − 5 = 36 − 5

Addition Property of Equality

If a, b, and c are any real numbers, and 𝑎 = 𝑏, then

𝒂 + 𝒄 = 𝒃 + 𝒄 and 𝒄 + 𝒂 = 𝒄 + 𝒃

If the same number is added to equal numbers, the sums are equal.

Subtraction Property of Equality

If a, b, and c are any real numbers, and 𝑎 = 𝑏, then

𝒂 − 𝒄 = 𝒃 − 𝒄

If the same number is subtracted from equal numbers, the differences are equal.

Subtraction Property of Equality

The subtraction property is just a special case of the addition property of equality, since subtracting the number c is the same as adding c. The addition property of equality guarantees that if 𝑎 = 𝑏, then

𝑎 + −𝑐 = 𝑏 + −𝑐

or 𝑎 − 𝑏 = 𝑎 − 𝑐

Solving Equations

The following examples show how to use the addition and subtraction properties of equality to solve some equations. You add the same number to, or subtract the same number from, each side of the equation in order to get an equation with the variable alone on one side of the equation.

Example 1

Solve 𝑥 − 8 = 17.

Solution

𝑥 − 8 = 17 Copy the equation.

Add 8 to each side.

𝑥 = 25 Simplify.

𝑥 − 8 + 8 = 17 + 8

Example 1

Solve 𝑥 − 8 = 17.

Check

𝑥 − 8 = 17 Copy the equation.

Substitute 25 for x.

17 = 17

25 − 8 = 17

The solution set is {25}.

The properties of real numbers guarantee in Example 1 that if the original equation, 𝑥 − 8 = 17, is true for some value of x, then the final equation, 𝑥 = 25, is also true for that value of x, and vice versa. Therefore, the two equations have the same solution set, {25}.

Example 2

Solve −5 = 𝑛 + 13.

Solution

−5 = 𝑛 + 13 Copy the equation.

Subtract 13 from each side.

−18 = 𝑛 Simplify.

−5 − 13 = 𝑛 + 13 − 13

Example 2

Solve −5 = 𝑛 + 13.

Check

−5 = 𝑛 + 13 Copy the equation.

Substitute 18 for n.

−5 = −5

−5 = −18 + 13

The solution set is {18}.

Equivalent Equations

Equations having the same solution set over a givendomain are called equivalent

equations.

To solve an equation you usuallychange, or transform, it into a simple equivalent equation whose solution set iseasy to see.

Solving Equations

To solve an equation, transform it to an equation that has the variable isolated on one side of the equal sign.

Transform, 𝑥 + 12 = 47

into 𝑥 = 35.

Therefore, the solution set is, {35}.

Transformations That Produce Equivalent Equations

1. Transforming by Substitution Substitute an equivalent expression for any expression in the given equation.

2. Transformation by Addition Add the same real number to each side of the given equation.

3. Transformation by Subtraction Subtract the same real number from each side of the given equation.

Class work

Oral Exercises

p 97: 1-21

Homework p 97: 1-39 odd, p 98: prob. 1-9 odd, p 98:40-48 even, p 99: prob. 10, 12, 13, p 100: MR