Example 1 Solving Equations with Variables on Both

Example 1: Solving Equations with Variables on Both Sides Solve 7 n – 2 = 5 n + 6 – 5 n 2 n – 2 = +2 2 n = To collect the variable terms on one side, subtract 5 n from both sides. 6 +2 8 Since n is multiplied by 2, divide both sides by 2 to undo the multiplication. n=4

Example 2: Solving Equations with Variables on Both Sides Solve 3 b + 2 = 4 b – 4 b -b + 2 = 0 – 2 -b = – 2 -1 -1 b =2 To collect the variable terms on one side, subtract 4 b from both sides. **If you end with a negative variable, you will need to multiply OR divide BOTH sides by -1 to make the variable positive. We want to know what the variable is; NOT the variable’s opposite.

Example 3: Solving Equations with Variables on Both Sides Infinite/Many Solutions Solve 10 – 5 x + 1 = 7 x + 11 – 12 x Identify like terms. 11 – 5 x = 11 – 5 x + 5 x 11 = 11 Combine like terms on the left and the right. Add 5 x to both sides. True statement. The equation 10 – 5 x + 1 = 7 x + 11 – 12 x is an identity. All values of x will make the equation true. All real numbers are solutions.

Identities and Contradictions WORDS Identity When solving an equation, if you get an equation that is always true, the original equation is an identity, and it has infinitely many solutions. NUMBERS 2+1=2+1 3=3 ALGEBRA 2+x=2+x –x –x 2=2

Example 4: Solving Equations with Variables on Both Sides No Solution Solve 12 x – 3 + x = 5 x – 4 + 8 x Identify like terms. 13 x – 3 = 13 x – 4 Combine like terms on the left and the right. Subtract 13 x from both sides. – 13 x – 3 = – 4 False statement. The equation 12 x – 3 + x = 5 x – 4 + 8 x is a contradiction. There is no value of x that will make the equation true. There are no solutions.

Identities and Contradictions WORDS Contradiction When solving an equation, if you get a false equation, the original equation is a contradiction, and it has no solutions. NUMBERS 1=1+2 1=3 ALGEBRA x= x+3 –x –x 0=3

Example 5: Solving Equations with Variables on Both Sides Solve 4 y + 7 – y = 10 + 3 y Identify like terms. 3 y + 7 = 3 y + 10 Combine like terms on the left and the right. Subtract 3 y from both sides. – 3 y 7 = 10 False statement. The equation 4 y + 7 – y = 10 + 3 y is a contradiction. There is no value of y that will make the equation true. There are no solutions.

Example 6: Solving Equations with Variables on Both Sides Solve 2 c + 7 + c = – 14 + 3 c + 21 Identify like terms. 3 c + 7 = 3 c + 7 – 3 c 7= Combine like terms on the left and the right. Subtract 3 c both sides. 7 True statement. The equation 2 c + 7 + c = – 14 + 3 c + 21 is an identity. All values of c will make the equation true. All real numbers are solutions.