MTH 091

Sections 3.2 and 3.3Solving Linear Equations

The Big Ideas• An equation is an statement that says two

algebraic expressions are equal.• Linear equations can be written in the form

Ax + B = C, where A, B, and C are real numbers and A is not equal to 0.

• The goal of solving a linear equation is to isolate the variable, to get x by itself on one side of the equation.

• The answer to the problem, or solution, can be substituted back into the equation to make a true sentence.

The Addition Property of Equality

• If A = B, then A + C = B + C• In words: we can add the same number to

each side of an equation without changing the equation.

• Connection to solving equations: if something is being added to/subtracted from your variable term, add/subtract that thing to/from both sides of the equation.

The Multiplication Property of Equality

• If A = B, then AC = BC• In words: we can multiply each side of an equation by

the same (non-zero) number without changing the solution.

• Connection to solving equations: if your isolated variable has a coefficient, divide both sides of the equation by that coefficient.

• If your isolated variable is being divided by a number, multiply both sides of the equation by that number.

• If the coefficient of your isolated variable is a fraction, multiply both sides of the equation by the reciprocal of that fraction.

Putting It All Together

1. Simplify each side separately.a. Got parentheses? Apply the distributive property.b. Got fractions? Multiply fractional coefficients and constants by the LCD.c. Got decimals? Multiply decimal coefficients and constants by 10, 100, 1000, etc.d. Combine like terms on each side of the equation. Do NOT combine like terms across the equal sign!

Continued…2. Isolate the variable term on one side.

a. If you have variable terms on the left side, leave them there. If you have variable terms on the right side,

move them to the left side (change sides, change signs).b. If you have constant terms on the right

side, leave them there. If you have constant terms on the left side, move them to the

right side (change sides, change signs).c. You can also do this by adding or subtracting terms

from both sides as necessary.d. Combine like terms on each side.

Almost Finished…

3. Isolate the variablea. You should now have a variable term on the left and a constant term on the right.b. Divide both sides of the equation by the

coefficient of the constant term.c. Your variable should now be isolated.

Examples

xx

xx

z

yy

x

9)54(2

14923

6713

536

832

More Examples

)7(645

0)4(4

122715)2(9

7)27(3

)54(521

yy

h

x

cc

kk

Something Unusual

• Sometimes, in trying to isolate your variable, that variable will “disappear”

• When this happens, there are two possible results:1. Your equation has no solution (contradiction)

because what remains is a false statement.2. Your equation has infinitely many solutions (identity)

because what remains is a true statement.

Examples

17202)34(5

102)5(23

23

)12(7714

yy

xx

xx

xx

Closing Remarks

• Many students already have strategies in place for solving linear equations. Whatever methods you choose to use:

• Practice• Practice• More Practice

Translating Phrases and Sentences to Algebraic Expressions and Equations• Look for words that indicate a particular

operation:“sum” or “more than” mean to add“difference”, “minus”, and “less than” mean to

subtract“product” and “times” mean to multiply“quotient” and “divided by” mean to divide“is” means equals • Where two types of words are used together,

parentheses may be necessary.

Still More Examples

Change each word phrase to an algebraic expression. Use x as the variable to represent the number.•9 subtracted from a number•Three added to the product of seven and a numberWrite each sentence as an equation.•The product of -5 and 9 gives -45.•The sum of -37 and 18 is -19.