SOLVING ONE-VARIABLE EQUATIONS •. Goal: Find the one value SOLVING ONE-VARIABLE EQUATIONS • Goal: Find the one value of the variable that makes the sentence true.
•. We can solve equations by. doing the OPPOSITE of what • We can solve equations by doing the OPPOSITE of what has been done to the variable in the problem. • If a problem says +, you subtract. • If a problem has multiplication, you divide.
By doing the opposite, we keep the sides of the equation balanced.
5x – 13 = 52
5x – 13 = 52 +13 +13 5x = 65
5x – 13 = 52 +13 +13 5x = 65 5 5 x = 13
12x + 1794 = 2127
12x + 1794 = 2127 x = 27.75
963 – 25x = 704
963 – 25x = 704 x = 10.36
What about this? 𝑦 5 −13=2
What about this? 𝑦 5 −13=2 Fractions mean division, so to cancel, we’ll add 13 and then multiply by 5. n = 75
Things that can complicate solving equations …
Parentheses •. Use distributive property. first. Like terms • Parentheses • Use distributive property first. Like terms • Combine them first.
4(3x – 7) = 48
4(3x – 7) = 48 12x – 28 = 48
4(3x – 7) = 48 12x – 28 = 48 12x = 76 x = 6.333…
-7(2x – 11) = 98
-7(2x – 11) = 98 -14x + 77 = 98 -14x = 21 x = -3/2 or -1.5
4p + 3 – 2p + 7 + 5p + 2 = 17
4p + 3 – 2p + 7 + 5p + 2 = 17 7p + 12 = 17 7p = 5 p = 5/7
5(3x + 5) – 3(2x – 1) = 145
5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145
5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145 9x + 28 = 145
5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145 9x + 28 = 145 9x = 117 x = 13
The goal is always to simplify The goal is always to simplify. Make the problem look like the easy ones we know how to solve.
Variable on Both Sides • Find the smaller number of the variable, and subtract that on both sides. • Solve the remaining problem.
5x – 15 = 2x + 72
5x – 15 = 2x + 72 -2x -2x 3x – 15 = 72
7x – 15 = 2x + 72 -2x -2x 3x – 15 = 72 3x = 87 x = 29
5x + 13 = 7x + 40
5x + 13 = 7x + 40 -5x -5x 13 = 2x + 40
5x + 13 = 7x + 40 -5x -5x 13 = 2x + 40 x = -27/2 or -13.5
3(2x + 7) = 3x + 4 + x + 9
3(2x + 7) = 3x + 4 + x + 9 6x + 21 = 4x + 13
3(2x + 7) = 3x + 4 + x + 9 6x + 21 = 4x + 13 2x + 21 = 13 2x = -8 x = -4
Special equations 2(3x – 7) = 6x + 11 10x – 15 = 5(2x – 3)
2(3x – 7) = 6x + 11 6x – 14 = 6x + 11 ?????
10x – 15 = 5(2x – 3) 10x – 15 = 10x – 15 ?????
When variables cancel out… •. If you have the exact When variables cancel out… • If you have the exact same thing on both sides (like 8 = 8), the answer is ALL REAL NUMBERS or INFINITELY MANY SOLUTIONS. 10x – 15 = 5(2x – 3) 10x – 15 = 10x – 15 -15 = -15
An equation with infinitely many solutions can also be called an IDENTITY.
•. If there is something. different on the 2 sides • If there is something different on the 2 sides (like 5 = 7), there is NO SOLUTION. 2(3x – 7) = 6x + 11 6x – 14 = 6x + 11 -14 = 11
•. If there is something. different on the 2 sides • If there is something different on the 2 sides (like 5 = 7), there is NO SOLUTION. 2(3x – 7) = 6x + 11 6x – 14 = 6x + 11 -14 = 11