Section 2.3-2.4 Solving Multi-Step and Variables on Both Sides Equations Algebra 1

Learning Targets Solve multi-step equations Write and solve multi-step equations in a context Solve consecutive integer problems Solve equations with variables on both sides Solve equations with special solutions (no solutions or many solutions) Write and solve an equation with variables on both sides

Solving Multi-Step Equations Purpose: Solve for a variable Procedure: Isolate the variable by performing the inverse operation. (Undoing/Going Backwards) Example: Solve 1. To “undo ”, we need to “” to both sides 2. To “undo multiplying by 11”, we need to “divide by 11 to both sides.” REMEMBER: Only when you’re moving from one side of the equal sign to the other!

2. 𝑎=5 (divide 2 to both sides) Example 1: Solve 2𝑎−6=4 1. 2𝑎=10 (add 6 to both sides) 2. 𝑎=5 (divide 2 to both sides)

Example 2 1. 𝑎+7=40 (multiply by 8 to both sides) SOLVE: 1. 𝑎+7=40 (multiply by 8 to both sides) 2. 𝑎=33 (subtract 7 from both sides)

Example 3 1. 2 3 𝑥−4 =6 (subtract 7 from both sides) 2. 2 𝑥−4 =18 (multiply by 3) 3. 2𝑥−8=18 (distribute 2) 4. 2𝑥=26 (add 8 to both sides) 5. 𝑥=13 (divide by 2) This is not the only way to solve this. Can you think of another way?

Example 4: Context Hiroshi is buying a pair of water skis that are on sale for 2 3 of the original price. After he uses a $25 gift card, the total cost before taxes is $115. What was the original price of the skis? 1. Price of water skis (W): ? 2. Sale for 2 3 of price: 2 3 𝑤 3. 2 3 𝑤−25=115 4. 2 3 𝑤=140 (add 25 to both sides) 5. 2𝑤=420 (multiple 3 to both sides) 6. 𝑤=$210 (divide by 2 to both sides)

Example 5: Context 1. # of CDs (H): ? 2. were sold: 3. 4. (add 10 to both sides) 5. 120 CDs (multiply by ) A music store sold 3 5 of their hip-hop CDs, but 10 were returned. Now the store has 62 CDs. How many were there originally?

Consecutive Integers These are numbers that are in counting order. 2 4 Example: 3 are consecutive numbers

Example 6: Consecutive Integers Find three consecutive odd numbers with a sum of -51. 1. Odd numbers are every other number. Let’s just start with the number 𝑛. 2. 3 consecutive odd numbers (like 3, 5, 7) would look like 𝑛, 𝑛+2, 𝑛+2+2 3. Thus, 𝑛+ 𝑛+2 + 𝑛+4 =−51 4. 3𝑛+6=−51 (combine like terms) 5. 3𝑛=−57 (subtract 6) 6. 𝑛=−19 (divide by 3) 7. So the numbers are: -19, -17, -15

Example 7: Variables on Both Sides Solve 2+5𝑘=3𝑘−6 1. Decide which side will be variables and which will be constants. (Left: Variable, Right: Constants) 2. Manipulate piece by piece 3. 2+2𝑘=−6 (subtract 3k) 4. 2𝑘=−8 (subtract 2) 5. 𝑘=−4 (divide by 2)

Example 8

Solve the equation 5𝑥+5=3 5𝑥−4 −10𝑥 Example 9 Solve the equation 5𝑥+5=3 5𝑥−4 −10𝑥 1. 5𝑥+5=15𝑥−12−10𝑥 2. 5𝑥+5=5𝑥−12 3. 5=−12 This is never true, so the answer is no solutions

Example 10 Solve the equation 3 2𝑏−1 −7=6𝑏−10 1. 6𝑏−3−7=6𝑏−10 2. 6𝑏−10=6𝑏−10 3. −10=−10 This is always true. Thus, we have many solutions or all real numbers as the answer.

Homework 2.3 & 2.4 Practice Worksheet Yes…there is a back!  Make sure to flip it over. Due MONDAY! You may ask 3 problems for tomorrow’s warm up!