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Checking Solutions to Equations

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Evaluating Algebraic Expressions and Equations

Your summer landscaping job pays a fixed rate of $20 per job plus $4 an hour. How much total money would you would make if it takes you 3 hours to complete a single job?

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Khan Academy: Variables Expressions and Equations

Guidance

You have probably seen letters in a mathematical expression, such as 3x-8 . These letters, also called variables, represent an unknown number. One of the goals of algebra is to solve various equations for a variable. Typically, x is used to represent the unknown number, but any letter can be used.

To evaluate an expression or equation, we would need to substitute in a given value for the variable and test it. In order for the given value to be true for an equation, the two sides of the equation must simplify to the same number.

Example A

Evaluate 2x^2-9 for x=-3 .

Solution: We know that 2x^2-9 is an expression because it does not have an equals sign. Therefore, to evaluate this expression, plug in -3 for x and simplify using the Order of Operations.

&2(-3)^2-9 \rightarrow (-3)^2=-3 \cdot -3=9\\&2(9)-9\\&18-9\\&9

You will need to remember that when squaring a negative number, the answer will always be positive. There are three different ways to write multiplication: 2 \times 9, 2 \cdot 9 , and 2(9).

Example B

Determine if x=5 is a solution to 3x-11=14 .

Solution: Even though the directions are different, this problem is almost identical to Example A. However, this is an equation because of the equals sign. Both sides of an equation must be equal to each other in order for it to be true. Plug in 5 everywhere there is an x . Then, determine if both sides are the same.

& \ ?\\3(5)-11 &= 14\\15-11 & \neq 14\\4 &\neq 14

Because 4 \ne 14 , this is not a true equation. Therefore, 5 is not a solution.

Example C

Determine if t=-2 is a solution to 7t^2-9t-10=36 .

Solution: Here, t is the variable and it is listed twice in this equation. Plug in -2 everywhere there is a t and simplify.

& \ ?\\7(-2)^2-9(-2)-10 &= 36\\& \ ?\\7(4)+18-10 &=36\\& \ ?\\28+18-10 &= 36\\36 &= 36

-2 is a solution to this equation.

Intro Problem Revisit Rewrite the sentence as an algebraic expression. $20 plus $4 an hour, would be 20+4h , where h equals the number of hours you work. Then, evaluate the expression for h = 3 .

20+4(3)=20+12=32

You will make a total of $32 for this particular job.

Guided Practice

1. Evaluate s^3-5s+6 for s=4 .

2. Determine if a=-1 is a solution to 4a-a^2+11=-2-2a .

Answers

1. Plug in 4 everywhere there is an s .

&4^3-5(4)+6\\&64-20+6\\&50

2. Plug in -1 for a and see if both sides of the equation are the same.

& \ ? \\4(-1)-(-1)^2+11 &= -2-2(-1)\\& \ ? \\-4-1+11 &=-2+2\\6 & \neq 0

Because the two sides are not equal, -1 is not a solution to this equation.

Vocabulary

Variable
A letter used to represent an unknown value.
Expression
A group of variables, numbers, and operators.
Equation
Two expressions joined by an equal sign.
Solution
A numeric value that makes an equation true.

Practice

Evaluate the following expressions for x = 5 .

  1. 4x-11
  2. x^2+8
  3. \frac{1}{2}x+1

Evaluate the following expressions for the given value.

  1. -2a+7; a=-1
  2. 3t^2-4t+5; t=4
  3. \frac{2}{3}c-7; c=-9
  4. x^2-5x+6; x=3
  5. 8p^2-3p-15; p=-2
  6. m^3-1; m=1

Determine if the given values are solutions to the equations below.

  1. x^2-5x+4=0; x=4
  2. y^3-7=y+3; x=2
  3. 7x-3=4; x=1
  4. 6z+z-5=2z+12;z=-3
  5. 2b-5b^2+1=b^2;b=6
  6. -\frac{1}{4}g+9=g+15;g=-8

Find the value of each expression, given that a=-1,b=2,c=-4 , and d=0 .

  1. ab-c
  2. b^2+2d
  3. c+\frac{1}{2}b-a
  4. b(a+c)-d^2

For problems 20-25, use the equation y^2+y-12=0 .

  1. Is y = 4 a solution to this equation?
  2. Is y = -4 a solution to this equation?
  3. Is y = 3 a solution to this equation?
  4. Is y = -3 a solution to this equation?
  5. Do you think there are any other solutions to this equation, other than the ones found above?
  6. Challenge Using the solutions you found from problems 20-23, find the sum of these solutions and their product. What do you notice?

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