Call Now: (800) 537-1660
 Home    Why?     Free Solver   Testimonials    FAQs    Product    Contact

May 21st

May 21st

# Lecture Notes for Math 097 Lesson 4

Algebra is the study of using symbols to express mathematical ideas. Solving linear 1
equations is the topic of this lesson.

 Example: The algebraic expression describes the population of the United States (in millions) x years after 1960. According to this model, when will the population reach 300 million?

Solving Equations

When we use the "=" symbol we are working with an equation. We can think of an
equation like a balance where both sides must be equal. We can add the same amount to
both sides and keep the equation balanced. We can divide both sides by the same number
(except zero) and keep the equation balanced. Our goal is to isolate, or "solve" for the
variable.

 Example solve for x answer x - 3 = 5 add 3 to both sides to undo subtracting 3 from x x - 3 + 3 = 5 + 3 x = 8

When solving linear equations, we are guaranteed that the solution to our final equation is
exactly the same as the solution to the original equation when we add, subtract, multiply
or divide both sides by the same number (except zero).

 Example solve for x answer 5x = -20 divide both sides by 5 to undo multiplying x by 5 5x / 5 = -20 / 5 x = -4

When solving linear equations, undo the operations that are being d one to x , to isolate x
on one side of the equation. If x is being multiplied by a number and also a number is
being added or subtracted, undo the addition or subtraction first to avoid fractions .

 Example solve for x answer -4x + 7 = -5 subtract 7 from both sides to undo adding 7 -4x + 7 - 7 = -5 - 7 -4x = -12 divide both sides by -4 to undo multiplying by -4 -4x / (-4) = -12 /(-4) x = 3

Suggested Steps in Solving Linear Equations

1. Simplify both sides of the equation (by removing parentheses and combining like
terms).
2. Use addition and subtraction to move all of the x terms to one side of the
equation, constant terms to the other.
3. Divide both sides by the coefficient of x to isolate x.

 Example solve for x answer -2( -3x - 1) = 5x - 1 6 x + 2 = 5x - 1 6x - 5x + 2 = 5x - 5x -1 x + 2 = -1 x + 2 - 2 = -1 - 2 x = -3 simplify left hand side subtract 5x from both sides subtract 2 from both sides

 Example: Is x = 0 a solution to the equation ? Answer No, because when you substitute x =0 into the left hand side it (3/2) does not equal 4. Example Answer Is x = -3 a solution to -2( -3x - 1) = 5x - 1 ? Yes, because -2 ( -3(-3) -1) = 5(-3) -1 -2 (9 -1) = 5(-3) - 1 -2( 8 ) = -15 - 1 - 16 = - 16

 Review A value for a variable is a solution to the equation, if when you substitute that value for the variable, both sides of the equation are equal

 Example solve for x answer 5x - 2x - 14 = 10 3x - 14 = 10 3x - 14 + 14 = 10 + 14 3x = 24 3x / 3 = 24 / 3 x = 8 simplify the left hand side add 14 to both sides divide both sides by 3 Example solve for x 3(5 - x) = 4(2x + 1) 15 - 3x = 8x + 4 15 - 3x + 3x=8x + 3x + 4 15 = 11x + 4 15 - 4 = 11x + 4 - 4 11 = 11x 11 /11 = 11x /11 x = 1 simplify both sides add 3x to both sides subtract 4 from both sides divide both sides by 11

Special Cases

Not all linear equations have one solution. Sometimes we begin with an identity where
every real number is a solution. Sometimes we begin with a false statement, where no
solution exists.

 Example solve for x answer 3( x - 2) = 3x - 6 any real number for x is a solution This equation is an identity, using the distributive property . Notice that when you complete the algebra, the variable drops out, and you are left with a true equation. 3x - 6 = 3x - 6 3x - 3x - 6 = 3x - 3x - 6 -6 = -6 subtract 3x from both sides Example solve for x answer 3( x - 2) = 3x - 2 there is no solution Notice that when you complete the algebra, the variable drops out, and you are left with a false equation. 3x - 6 = 3x - 5 3x - 3x - 6 = 3x - 3x - 5 -6 = -5 subtract 3x from both sides If you try any number for x in this false equation you will see that you will never get the same number on both sides of the equation.

For additional practice, try the worksheet below.

Worksheet
Solve for x

Worksheet