Math 1 · Lesson 5 · part 5 of 5
Practice Problems
Try them, get help one step at a time, then print how it went.
Warm-up: one move at a time
Four short ones. Each asks for a single move, not a whole solution. Say the property out loud as you type.
Practice: try it, then get help one step at a time
Try each problem yourself first. Hint gives a nudge, Show next step shows one worked step at a time, Reveal answer checks, and Hide help / start over clears the card. Every worked step names the property it uses. The problems climb from two like terms up to a full solution you have to name line by line.
My results
Finished the practice? Print a page that shows each question, what you typed or chose, the correct answer, and how it went. Good for showing a teacher or keeping in a folder.
Exit ticket
On paper or an index card. Use the math word and the plain-English word.
- Name the three steps in order. Which step do you distribute in?
- One of these has a division that Step 1 rewrites, and one does not: x ÷ 4 + 3 = 8, and x4 + 3 = 8. Say which is which, and give the coefficient (the number on the variable) on x in each one.
- Solve 3(x + 2) + 2x = x + 2. Show Step 1, name the property on every line, and check with plus the opposite.
- In 8x + 1 = 3x + 16, which x term would you move, what do you add to both sides, and which property lets you do that? Is there only one right choice?
Check your exit ticket
- Step 1 is Rewrite, which Lesson 3 called translating. It makes the line only addition and multiplication, with the invisible 1s written in. Step 2 is distribute, then combine like terms (terms with the exact same letter part). Step 3 puts the x terms on one side and the plain numbers on the other, then does one last multiplication. You distribute inside Step 2, at the start of it.
- The line x ÷ 4 + 3 = 8 has a written ÷ sign, so Step 1 rewrites it. Dividing by 4 is multiplying by 14, the multiplicative inverse (the reciprocal) of 4. So that line becomes 14 x + 3 = 8. The line x4 + 3 = 8 is different. A fraction bar is not a written ÷ sign, so Step 1 has nothing to rewrite there. x4 already is the coefficient (the number on the variable) 14 on x. Both lines end up with the same coefficient (the number on the variable), which is 14.
- Step 1 rewrite: there is no written ÷ sign and no subtraction here. Only the invisible 1s go in, so the line is 3(1x + 2) + 2x = 1x + 2. Step 2, distribute first: 3(1x + 2) = 3 × 1x + 3 × 2, by the distributive property (hand one out to everyone). Now 3 × 1x = (3 × 1)x, by the associative property of multiplication (the factors change partners, the grouping moves), so that term is 3x. And 3 × 2 = 6, so the line is 3x + 6 + 2x = 1x + 2. Step 2, combine: use the commutative property of addition (the terms trade places) to put 3x and 2x next to each other. Then use the associative property of addition (the terms change partners, the grouping moves) to group them. Then (3 + 2)x = 5x, by the distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2). The line is 5x + 6 = 1x + 2. Step 3: the coefficients (the numbers on the variables) are 5 and 1. We prefer to move the smaller one, so move the 1x on the right. Add −1x to both sides, the additive inverse (the opposite) of 1x, by the addition property of equality (add the same thing to both sides). On the right, 1x + (−1x) = (1 + (−1))x = 0x, by the distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2). The zero rule (0 times anything is 0) turns 0x into 0, and 0 + 2 = 2 by the additive identity (the do-nothing number 0). On the left, (5 + (−1))x = 4x, by the distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2) again. The line is 4x + 6 = 2. Add −6 to both sides, the additive inverse (the opposite) of 6, by the addition property of equality (add the same thing to both sides). Then 6 + (−6) = 0 by the additive identity (the do-nothing number 0), and 2 + (−6) = −4, so 4x = −4. Last move: multiply both sides by 14, the multiplicative inverse (the reciprocal) of the coefficient (the number on the variable) 4. That is the multiplication property of equality (multiply both sides by the same thing). On the left, 14 × 4x = (14 × 4)x, by the associative property of multiplication (the factors change partners, the grouping moves). Then 14 × 4 = 1, and 1x = x by the multiplicative identity (the do-nothing number 1). On the right, 14 × (−4) = −1. So x = −1. Check: 3 × ((−1) + 2) + 2 × (−1) = 3 × 1 + (−2) = 3 + (−2) = 1, and 1 × (−1) + 2 = (−1) + 2 = 1. Both sides are 1, so x = −1 is right.
- We prefer to move the 3x, because 3 is the smaller coefficient (the number on the variable). Then the x you keep has a positive coefficient (the number on the variable), and the arithmetic stays easier. That is a preference, not a rule. Moving the 8x instead is allowed, and it gives the same answer, x = 3. To move the 3x, add −3x to both sides, the additive inverse (the opposite) of 3x. The addition property of equality (add the same thing to both sides) lets you do that, and Lesson 4 called it doing the same thing to both sides. Then 3x + (−3x) = (3 + (−3))x = 0x, by the distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2). The zero rule (0 times anything is 0) turns 0x into 0, and 0 + 16 = 16 by the additive identity (the do-nothing number 0). So no x is left on the right side.