Warm Up # 70

Solving Two-Step Equations CCSS Aligned My mom warned me that if I hold sneezes in, my head might grow abnormally large! Only those with single-step equation experience dare proceed! ©Mike’s Math Mall

Equation Solving When we solve equations, our goal is to find the value of the variable; the solution to the equation. 𝟏𝟒𝒙−𝟑𝟕=𝟏𝟕𝟑 I’m afraid we’ll need a time machine and travel back to around 210 AD in order to do that, Sparky! My goal is to find the person responsible for mixing letters in with math! I’ll pack my bags!

Equation Solving To find the solution to an equation, we isolate the variable by mathematically undoing the equation. If you can name a two-step equation, you will automatically know how to undo or solve it. Check this out! →

Two-Step Equations 𝟏𝟒𝒙−𝟑𝟕=𝟏𝟕𝟑 There are four types of two-step equations: Multiplication/Addition Division/Addition Multiplication/Subtraction Division/Subtraction What type of an equation is this? 𝟏𝟒𝒙−𝟑𝟕=𝟏𝟕𝟑 If you said multiplication/subtraction, you’re correct!

Name that Equation! 1) 𝒙 𝟗 +𝟓=𝟏𝟎 2) 𝟕=𝟔𝒚−𝟏𝟏 3) 𝟒𝒏+𝟖=𝟐𝟒 4) 𝟑= 𝒓 𝟐 −𝟏𝟕 Name the following equations. 1) 𝒙 𝟗 +𝟓=𝟏𝟎 Division/Addition 2) 𝟕=𝟔𝒚−𝟏𝟏 Multiplication/Subtraction 3) 𝟒𝒏+𝟖=𝟐𝟒 Multiplication/Addition 4) 𝟑= 𝒓 𝟐 −𝟏𝟕 Division/Subtraction

Two-Step Equations Because we use inverse operations to solve equations, we just use the opposite of an equation’s name to solve it. Great question, Sparticus! You MUST undo the addition or subtraction part first! With just one catch! Great Caesar's ghost! Why does there ALWAYS have to be a catch? How does naming equations help you solve them? Must… find…happy place…now! Calm down and I’ll show you what I’m talking about!

Solving Two-Step Equations Let’s start with this equation: 𝟕𝒏+𝟒=𝟓𝟑 Step 1: Name the equation. Multiplication/Addition Step 2: Name the inverse. Division/Subtraction Continued →

Solving Two-Step Equations Step 3: Undo the addition part first (with subtraction) 𝟕𝒏+𝟒=𝟓𝟑 − 𝟒 −𝟒 𝟕𝒏=𝟒𝟗 Step 4: Undo multiplication part next (with division) 𝟕𝒏=𝟒𝟗 𝒏=𝟕 𝟕 𝟕

Don’t Forget to  the Solution 𝒏=𝟕 𝟕𝒏+𝟒=𝟓𝟑 𝟕(𝟕)+𝟒=𝟓𝟑 𝟒𝟗+𝟒=𝟓𝟑 

Solving Two-Step Equations Let’s try another one: 𝒙 𝟖 −𝟔=𝟓 Step 1: Name the equation. Division/Subtraction Step 2: Name the inverse. Multiplication/Addition Continued →

Solving Two-Step Equations Step 3: Undo the subtraction part first (with addition) 𝒙 𝟖 −𝟔=𝟓 + 𝟔 +𝟔 𝒙 𝟖 =𝟏𝟏 Step 4: Undo the division part next (with multiplication) 𝟖 𝟏 𝒙 𝟖 =𝟏𝟏 ∙ ∙ 𝟖 𝒙=𝟖𝟖

Don’t Forget to  the Solution 𝒙=𝟖𝟖 To be honest, that’s not what I’m finding a little weird right now! 𝒙 𝟖 −𝟔=𝟓 Is it a little weird that I’m actually starting to understand this stuff? I don’t get it! Why is everyone laughing? 𝟖𝟖 𝟖 −𝟔=𝟓 𝟏𝟏−𝟔=𝟓 

Let’s Try a Couple More! 𝟏𝟎𝒚−𝟏𝟏=𝟔𝟗 𝟏𝟎𝒚−𝟏𝟏=𝟔𝟗 𝟏𝟎𝒚=𝟖𝟎 + 𝟏𝟏 + 𝟏𝟏 Step 1: Name it! Multiplication/Subtraction Step 2: Think inverse! Division/Addition Step 3: Undo Subtraction first! 𝟏𝟎𝒚−𝟏𝟏=𝟔𝟗 + 𝟏𝟏 + 𝟏𝟏 𝟏𝟎𝒚=𝟖𝟎 Continued →

Let’s Try a Couple More! 𝒚=𝟖 𝟏𝟎𝒚=𝟖𝟎 𝟏𝟎𝒚−𝟏𝟏=𝟔𝟗 𝟏𝟎(𝟖)−𝟏𝟏=𝟔𝟗 𝟖𝟎−𝟏𝟏=𝟔𝟗  Step 4: Undo Multiplication! 𝟏𝟎𝒚=𝟖𝟎 𝟏𝟎 𝟏𝟎 𝒚=𝟖 Check it! 𝟏𝟎𝒚−𝟏𝟏=𝟔𝟗 𝟏𝟎(𝟖)−𝟏𝟏=𝟔𝟗 𝟖𝟎−𝟏𝟏=𝟔𝟗 

Last One! 𝟏𝟖= 𝒏 𝟕 +𝟗 𝟏𝟖= 𝒏 𝟕 +𝟗 𝟗= 𝒏 𝟕 −𝟗 −𝟗 Continued → Step 1: Name it! Division/Addition Step 2: Think inverse! Multiplication/Subtraction Step 3: Undo Addition first! 𝟏𝟖= 𝒏 𝟕 +𝟗 −𝟗 −𝟗 𝟗= 𝒏 𝟕 Continued →

Last One! ∙ ∙ 𝟗= 𝒏 𝟕 𝟕 𝟏 𝟕 or 𝟔𝟑=𝒏 𝒏=𝟔𝟑 𝟏𝟖= 𝒏 𝟕 +𝟗 𝟏𝟖= 𝟔𝟑 𝟕 +𝟗 Step 4: Undo Division! 𝟗= 𝒏 𝟕 𝟕 𝟏 𝟕 ∙ ∙ 𝟔𝟑=𝒏 or 𝒏=𝟔𝟑 Check it! 𝟏𝟖= 𝒏 𝟕 +𝟗 𝟏𝟖= 𝟔𝟑 𝟕 +𝟗 𝟏𝟖=𝟗+𝟗 

It’s Time to Show Your Stuff! Solve the following equations. Be sure to include the variable with your answer. 2) 𝒎 𝟐 +𝟕=𝟏𝟗 1) 𝟑𝒙−𝟗=𝟏𝟖 𝒙=𝟗 𝒎=𝟐𝟒 3) 𝒏 𝟓 −𝟐𝟎=𝟓 4) 𝟔𝒓+𝟏𝟒=𝟖𝟔 𝒏=𝟏𝟐𝟓 𝒓=𝟏𝟐 6) 𝟐𝟐= 𝒚 𝟏𝟓 +𝟏𝟕 5) 𝟏𝟎𝟒=𝟖𝒙−𝟖 𝒙=𝟏𝟒 𝒚=𝟕𝟓

Let’s Try it with some Integers! Solve the following equations. Be sure to include the variable with your answer. 2) 𝒚 𝟔 −𝟏=−𝟖 1) −𝟒𝒙+𝟏𝟏=−𝟐𝟏 𝒙=𝟖 𝒚=−𝟒𝟐 3) − 𝒄 𝟓 +𝟏𝟐=𝟕 4) 𝟓𝒋−𝟐𝟏=−𝟓𝟔 𝒄=𝟐𝟓 𝒋=−𝟕 6) −𝟏𝟒=− 𝒂 𝟖 +𝟗 5) 𝟏𝟐𝟐=−𝟔𝒃+𝟐𝟔 𝒃=−𝟏𝟔 𝒂=𝟏𝟖𝟒

Equations and Word Problems! The perimeter of a rectangle is 54 inches. Its length is 22 inches. Write and solve an equation to determine its width? Perimeter = 54 inches width 22 in Start by defining a variable! Let 𝒘 = width 2 times the width + 2 times the length = perimeter 𝟐𝒘+𝟒𝟒=𝟓𝟒 𝟐𝒘=𝟏𝟎 −𝟒𝟒 −𝟒𝟒 𝒘=𝟓 𝒊𝒏 𝟐 𝟐 𝟐𝒘=𝟏𝟎

Equations and Word Problems! Katie wants to buy a motor scooter that costs $518. She has already saved $188 and will earn $22 each week from babysitting jobs. Write and solve an equation to see how many weeks it will take Katie to save enough money to buy the motor scooter. Define a variable! Let 𝒘 = number of weeks amount saved each week + amount already saved = total cost 𝟐𝟐𝒘=𝟑𝟑𝟎 𝟐𝟐𝒘+𝟏𝟖𝟖=𝟓𝟏𝟖 −𝟏𝟖𝟖 −𝟏𝟖𝟖 𝟐𝟐 𝟐𝟐 𝟐𝟐𝒘=𝟑𝟑𝟎 𝒘=𝟏𝟓 𝒘𝒆𝒆𝒌𝒔

Equations and Word Problems! Let 𝒂 = weekly allowance Ashley spent half of her weekly allowance at the movies. To earn extra money, her parents let her clean the house windows for $8. What is Ashley’s weekly allowance if she ended up with $14? Define a variable! Let 𝒂 = weekly allowance amount of allowance spent at the movies + amount earned washing windows = ending amount of money 𝒂 𝟐 +𝟖=𝟏𝟒 𝒂 𝟐 =𝟔 𝟐 ∙ ∙ To find “half” of something, divide by 2. 𝟐 −𝟖 −𝟖 𝒂 𝟐 =𝟔 𝒂=$𝟏𝟐

Solving Two-Step Equations We could start by getting you out of that baby swing! Then it looks like my work here is done! So, how did you do on the practice problems, Sparky? I’m glad I could help! Now is there any chance you can give me some pointers on how to impress the ladies? I did great with your help, sir! Knowing multiplication and division facts also made things easier. Baby swing? This is a loves seat built just for me and that special “Mrs. Sparkles!” ©Mike’s Math Mall On your way out, tell the DJ to put on a slow one, will ya?