1. Solving f(x) = g(x) Algebraically
Unit 4 Lesson 5 Notes 2
Solving f(x) = g(x) Algebraically

2. Example:
Barbara has a bunny that weighs 5 pounds and gains 3 pounds per year. Her cat weighs 19 pounds and gains 1 pound per year. When will the bunny and the cat weigh the same amount?
If x is the number of years and y is the weight, write an equation that represents the bunny’s weight (y) after a given number of years (x).
 Bunny: y = 3x + 5

3. Barbara has a bunny that weighs 5 pounds and gains 3 pounds per year
Barbara has a bunny that weighs 5 pounds and gains 3 pounds per year. Her cat weighs 19 pounds and gains 1 pound per year. When will the bunny and the cat weigh the same amount?
If x is the number of years and y is the weight, write an equation that represents the cat’s weight (y) after a given number of years (x).
 Cat: y = 1x + 19

4. The Bunny and Cat’s weight will be equal after 7 years.
When two equations are equal to the same thing
(or variable) then you can put them equal to each other and solve.
-1x x
2x + 5 = 19
2x = 14
x = 7
Bunny: y = 3x + 5
Cat: y = x + 19
3x + 5 =
x + 19
The Bunny and Cat’s weight will be equal after 7 years.

5. Solving f(x) = g(x) Algebraically
Replace any function notation
f(x), g(x), with y =
Place the two functions equal to each other.
Looking at what type of function you have, solve the equation using appropriate method.

6. Solving Different types of Equations
Quadratic:
If there is an x2 and NO x term, get x2 by itself and take square root of both sides.
If there is an x2 AND an x term, get one side equal to zero, factor , put equal to zero and then solve for x.
Absolute Value:
Get |x| by itself on one side of equal sign.
Make two equations and set equal to positive answer AND negative answer.
Solve for x.
Linear:
Get x to one side of equal sign.
If a constant is with x, add/subtract that value to both sides.
If a number is being multiplied by x, divide both sides by that number.

7. Example 1: Find the solution to f(x) = g(x).
f(x) = 7x – 3 and g(x) = x + 15
(same as y = 7x – 3 and y = x + 15)
7x – 3 = x + 17
6x – 3 = 15
6x = 18
x = 3
Both are linear functions

8. Example 2: Find the solution to f(x) = g(x).
f(x) = x2 – 4 and g(x) = 45 x2 – 4 = 45 x2 = 49 x = 7 and -7
Quadratic but NO ‘x’ term

9. Example 3: Find the solution to f(x) = g(x).
f(x) = x2 – 5x and g(x) = 2x – 10 x2 – 5x = 2x – 10 x2 – 7x + 10 = 0 (x – 5)(x – 2) = 0 x = 5 and x = 2
Quadratic - with x2 AND ‘x’ term

10. Example 4: Find the solution to f(x) = g(x).
f(x) = |x – 19| and g(x) = 22
|x – 19| = 22
x – 19 = x – 19 = -22
x = and x = -3
Absolute Value Function