Math 1 · Lesson 5 · part 3 of 5
Three worked examples, every line named
The Step 1 rewrite is shown in this color .
Three worked examples, every line named
The Step 1 rewrite is shown in this color. The number you add to both sides or multiply both sides by is shown in this color.
4x - 9 + x = 2x + 6
The original4x - 9 + x = 2x + 6There is a subtraction, a lonely x, and an x term on each side. The like terms (the terms with the same letter part) are not next to each other.
Step 1 rewrite4x + (−9) + 1x = 2x + 6Subtracting 9 is adding −9, its additive inverse (the opposite). Lesson 3 called this translating. The lonely x is really 1x, by the multiplicative identity (the do-nothing number 1). There is no written ÷ sign, so there is no division to rewrite.
Step 2 commute4x + 1x + (−9) = 2x + 6There are no parentheses, so there is nothing to distribute. The commutative property of addition (the terms trade places) brings the two like terms (the terms with the same letter part) next to each other. This is allowed because Step 1 made the line all addition.
Step 2 associate(4x + 1x) + (−9) = 2x + 6The associative property of addition (the terms change partners, the grouping moves) groups the two like terms (the terms with the same letter part). Now they can be combined.
Step 2 combine(4 + 1)x + (−9) = 2x + 6, so 5x + (−9) = 2x + 6The distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2) pulls the x out of both terms. Then 4 + 1 = 5. This is why Lessons 3 and 4 told you to add the coefficients (the numbers on the variables).
Step 3 move 2x5x + (−9) + (−2x) = 2x + 6 + (−2x)The coefficients (the numbers on the variables) are 5 and 2, so 2x is the smaller x term. Move that one. Add its additive inverse (the opposite), which is −2x, to both sides. That is the addition property of equality (add the same thing to both sides).
Step 3 combine(5 + (−2))x + (−9) = 0x + 6, so 3x + (−9) = 6On both sides the x terms come together first. That takes the commutative property of addition (the terms trade places) and the associative property of addition (the terms change partners, the grouping moves). On the left, 5x + (−2x) = (5 + (−2))x = 3x. On the right, 2x + (−2x) = (2 + (−2))x = 0x = 0. Both of those lines are the distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2). The right side also uses the zero rule (0 times anything is 0). Then 0 + 6 = 6, by the additive identity (the do-nothing number 0).
Step 3 opposite of −93x + (−9) + 9 = 6 + 9, so 3x + 0 = 15, that is 3x = 15The additive inverse (the opposite) of −9 is 9. Add it to both sides, by the addition property of equality (add the same thing to both sides). On the left, (−9) + 9 = 0, and 3x + 0 = 3x, by the additive identity (the do-nothing number 0). On the right, 6 + 9 = 15.
Step 3 reciprocal13 × 3x = 13 × 15, so (13 × 3)x = 5, so 1x = 5The multiplicative inverse (the reciprocal) of the coefficient (the number on the variable) 3 is 13. Multiply both sides by it, by the multiplication property of equality (multiply both sides by the same thing). On the left, the associative property of multiplication (the factors change partners, the grouping moves) regroups 13 × 3x into (13 × 3)x. Then 13 × 3 = 1, so the left side is 1x. On the right, 13 × 15 = 5.
Answerx = 51x = x, by the multiplicative identity (the do-nothing number 1). The opposite made 0 and the reciprocal made 1. Those are the undo partners from Lesson 1.
Check4 × 5 + (−9) + 1 × 5 = 20 + (−9) + 5 = 16, and 2 × 5 + 6 = 16. Both sides are 16, so x = 5 is right.Put 5 back into the original equation, written with plus the opposite. The lonely x is 1x, so it is 1 × 5. A check never uses a ÷ sign.
5(x - 2) + 2x = 3x + 6
The original5(x - 2) + 2x = 3x + 6There are parentheses with a subtraction inside them. There is also an x term on each side.
Step 1 rewrite5(1x + (−2)) + 2x = 3x + 6Subtracting 2 is adding −2, its additive inverse (the opposite). Lesson 3 called this translating. The x inside the parentheses is really 1x, by the multiplicative identity (the do-nothing number 1). There is no written ÷ sign, so there is no division to rewrite.
Step 2 distribute5 × 1x + 5 × (−2) + 2x = 3x + 6, so 5x + (−10) + 2x = 3x + 6The distributive property (hand one out to everyone) hands the 5 to every term (the pieces that are added) inside. Since 5 × 1 = 5, the first product is 5x. And 5 × (−2) = −10. Distributing over the subtraction directly is allowed too: 5(x - 2) = 5x - 10, as Lesson 2 shows.
Step 2 commute5x + 2x + (−10) = 3x + 6The commutative property of addition (the terms trade places) brings the two like terms (the terms with the same letter part) next to each other.
Step 2 associate(5x + 2x) + (−10) = 3x + 6The associative property of addition (the terms change partners, the grouping moves) groups them so they can be combined.
Step 2 combine(5 + 2)x + (−10) = 3x + 6, so 7x + (−10) = 3x + 6The distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2) pulls the x out of both terms. Then 5 + 2 = 7.
Step 3 move 3x7x + (−10) + (−3x) = 3x + 6 + (−3x)The coefficients (the numbers on the variables) are 7 and 3, so 3x is the smaller x term. Move that one. Add its additive inverse (the opposite), which is −3x, to both sides. That is the addition property of equality (add the same thing to both sides).
Step 3 combine(7 + (−3))x + (−10) = 0x + 6, so 4x + (−10) = 6On both sides the x terms come together first. That takes the commutative property of addition (the terms trade places) and the associative property of addition (the terms change partners, the grouping moves). On the left, 7x + (−3x) = (7 + (−3))x = 4x. On the right, 3x + (−3x) = (3 + (−3))x = 0x = 0. Both of those lines are the distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2). The right side also uses the zero rule (0 times anything is 0). Then 0 + 6 = 6, by the additive identity (the do-nothing number 0).
Step 3 opposite of −104x + (−10) + 10 = 6 + 10, so 4x + 0 = 16, that is 4x = 16The additive inverse (the opposite) of −10 is 10. Add it to both sides, by the addition property of equality (add the same thing to both sides). On the left, (−10) + 10 = 0, and 4x + 0 = 4x, by the additive identity (the do-nothing number 0). On the right, 6 + 10 = 16.
Step 3 reciprocal14 × 4x = 14 × 16, so (14 × 4)x = 4, so 1x = 4The multiplicative inverse (the reciprocal) of the coefficient (the number on the variable) 4 is 14. Multiply both sides by it, by the multiplication property of equality (multiply both sides by the same thing). On the left, the associative property of multiplication (the factors change partners, the grouping moves) regroups 14 × 4x into (14 × 4)x. Then 14 × 4 = 1, so the left side is 1x. On the right, 14 × 16 = 4.
Answerx = 41x = x, by the multiplicative identity (the do-nothing number 1). The opposite made 0 and the reciprocal made 1, the same undo partners as before.
Check5 × (4 + (−2)) + 2 × 4 = 5 × 2 + 8 = 10 + 8 = 18, and 3 × 4 + 6 = 18. Both sides are 18, so x = 4 is right.Put 4 back into the original equation, written with plus the opposite. Do the parentheses first, so 4 + (−2) = 2. A check never uses a ÷ sign.
x ÷ 3 + 2x + 5 = −9
The originalx ÷ 3 + 2x + 5 = −9This one has a written ÷ sign, so Step 1 really does have a division to rewrite. There is a lonely x hiding inside it. The answer will be negative, and a negative answer is a real answer.
Step 1 rewrite13 x + 2x + 5 = −9The lonely x is 1x, by the multiplicative identity (the do-nothing number 1). Dividing by 3 is multiplying by 13, the multiplicative inverse (the reciprocal) of 3. So 1x ÷ 3 becomes 13 x. Lesson 3 called this translating. If it had been written x3, Step 1 would have nothing to rewrite, because a fraction bar is not a written ÷ sign.
Step 2 associate(13 x + 2x) + 5 = −9There are no parentheses, so there is nothing to distribute. The two like terms (the terms with the same letter part) are already next to each other. The associative property of addition (the terms change partners, the grouping moves) groups them.
Step 2 combine(13 + 2)x + 5 = −9, so 73 x + 5 = −9The distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2) pulls the x out of both terms. Now add the coefficients (the numbers on the variables). Write 2 as 63, so 13 + 63 = 73. The coefficient (the number on the variable) on x is now 73.
Step 3 opposite of 573 x + 5 + (−5) = −9 + (−5), so 73 x + 0 = −14, that is 73 x = −14There is only one x term, so no x term has to move. The additive inverse (the opposite) of 5 is −5. Add it to both sides, by the addition property of equality (add the same thing to both sides). On the left, 5 + (−5) = 0, and 73 x + 0 = 73 x, by the additive identity (the do-nothing number 0). On the right, −9 + (−5) = −14.
Step 3 reciprocal37 × 73 x = 37 × (−14), so (37 × 73)x = −6, so 1x = −6The multiplicative inverse (the reciprocal) of the coefficient (the number on the variable) 73 is 37. You flip it over. Multiply both sides by it, by the multiplication property of equality (multiply both sides by the same thing). On the left, the associative property of multiplication (the factors change partners, the grouping moves) regroups 37 × 73 x into (37 × 73)x. Then 37 × 73 = 2121 = 1, so the left side is 1x. On the right, 17 × (−14) = −2, and 3 × (−2) = −6.
Answerx = −61x = x, by the multiplicative identity (the do-nothing number 1). Nothing special happens because the answer is negative. You check it the same way.
Check13 × (−6) + 2 × (−6) + 5 = (−2) + (−12) + 5 = −9, and the right side is −9. Both sides are −9, so x = −6 is right.Put −6 back in, using the Step 1 rewrite, because a check never uses a ÷ sign. That is why the first piece is 13 × (−6) and not a division.