Solved on Mar 18, 2024

Solve multi-step linear equations. 1. $7c - 3 + 2c = 15$ 2. $5(x - 7) = 20$ 3. $6h + 5 = 10 + h$ 4. $60 = -4(3y - 6)$ 5. $-5g + 24 = 7g - 24$ 6. $22 = 3u + 10 - 5u$ 7. $-5(3k - 3) + 17 = -43$ 8. $7t - 5 + 4t = 3t - 21$ 9. $5m - 7 = 3(2m + 1)$ 10. $-12 = 4(2q + 7) - 3q$ 11. $-11 - 6b + 3 = 16 + 2b$ 12. $5(8 - 2w) = 4 - 4w$

STEP 1

Assumptions
1. Each equation is a linear equation with one variable.
2. The equations are to be solved for the variable.
3. The solutions are real numbers.

STEP 2

Solve the first equation $7c - 3 + 2c = 15$ .
Combine like terms on the left side of the equation.
$7c + 2c - 3 = 15$

STEP 3

Add the coefficients of $c$ .
$9c - 3 = 15$

STEP 4

Add 3 to both sides of the equation to isolate the term with the variable.
$9c - 3 + 3 = 15 + 3$

STEP 5

Simplify both sides of the equation.
$9c = 18$

STEP 6

Divide both sides by 9 to solve for $c$ .
$c = \frac{18}{9}$

STEP 7

Calculate the value of $c$ .
$c = 2$

STEP 8

Solve the second equation $5(x - 7) = 20$ .
Distribute the 5 on the left side of the equation.
$5x - 35 = 20$

STEP 9

Add 35 to both sides of the equation to isolate the term with the variable.
$5x - 35 + 35 = 20 + 35$

STEP 10

Simplify both sides of the equation.
$5x = 55$

STEP 11

Divide both sides by 5 to solve for $x$ .
$x = \frac{55}{5}$

STEP 12

Calculate the value of $x$ .
$x = 11$

STEP 13

Solve the third equation $6h + 5 = 10 + h$ .
Subtract $h$ from both sides of the equation to get all the $h$ terms on one side.
$6h - h + 5 = 10$

STEP 14

Combine like terms.
$5h + 5 = 10$

STEP 15

Subtract 5 from both sides of the equation to isolate the term with the variable.
$5h + 5 - 5 = 10 - 5$

STEP 16

Simplify both sides of the equation.
$5h = 5$

STEP 17

Divide both sides by 5 to solve for $h$ .
$h = \frac{5}{5}$

STEP 18

Calculate the value of $h$ .
$h = 1$

STEP 19

Solve the fourth equation $60 = -4(3y - 6)$ .
Distribute the -4 on the right side of the equation.
$60 = -4 \cdot 3y + 4 \cdot 6$

STEP 20

Simplify the right side of the equation.
$60 = -12y + 24$

STEP 21

Subtract 24 from both sides of the equation to isolate the term with the variable.
$60 - 24 = -12y$

STEP 22

Simplify both sides of the equation.
$36 = -12y$

STEP 23

Divide both sides by -12 to solve for $y$ .
$y = \frac{36}{-12}$

STEP 24

Calculate the value of $y$ .
$y = -3$

STEP 25

Solve the fifth equation $-5g + 24 = 7g - 24$ .
Add $5g$ to both sides of the equation to get all the $g$ terms on one side.
$-5g + 5g + 24 = 7g + 5g - 24$

STEP 26

Combine like terms.
$24 = 12g - 24$

STEP 27

Add 24 to both sides of the equation to isolate the term with the variable.
$24 + 24 = 12g$

STEP 28

Simplify both sides of the equation.
$48 = 12g$

STEP 29

Divide both sides by 12 to solve for $g$ .
$g = \frac{48}{12}$

STEP 30

Calculate the value of $g$ .
$g = 4$
The solutions for the first five equations are:
1. $c = 2$
2. $x = 11$
3. $h = 1$
4. $y = -3$
5. $g = 4$

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STEP 1

Assumptions1. Each equation is a linear equation with one variable.2. The equations are to be solved for the variable.3. The solutions are real numbers.

STEP 2

Solve the first equation 7c−3+2c=157c - 3 + 2c = 157c−3+2c=15.Combine like terms on the left side of the equation.7c+2c−3=157c + 2c - 3 = 157c+2c−3=15

STEP 3

Add the coefficients of ccc.9c−3=159c - 3 = 159c−3=15

STEP 4

Add 3 to both sides of the equation to isolate the term with the variable.9c−3+3=15+39c - 3 + 3 = 15 + 39c−3+3=15+3

STEP 5

Simplify both sides of the equation.9c=189c = 189c=18

STEP 6

Divide both sides by 9 to solve for ccc.c=189c = \frac{18}{9}c=918​

STEP 7

Calculate the value of ccc.c=2c = 2c=2

STEP 8

Solve the second equation 5(x−7)=205(x - 7) = 205(x−7)=20.Distribute the 5 on the left side of the equation.5x−35=205x - 35 = 205x−35=20

STEP 9

Add 35 to both sides of the equation to isolate the term with the variable.5x−35+35=20+355x - 35 + 35 = 20 + 355x−35+35=20+35

STEP 10

Simplify both sides of the equation.5x=555x = 555x=55

STEP 11

Divide both sides by 5 to solve for xxx.x=555x = \frac{55}{5}x=555​

STEP 12

Calculate the value of xxx.x=11x = 11x=11

STEP 13

Solve the third equation 6h+5=10+h6h + 5 = 10 + h6h+5=10+h.Subtract hhh from both sides of the equation to get all the hhh terms on one side.6h−h+5=106h - h + 5 = 106h−h+5=10

STEP 14

Combine like terms.5h+5=105h + 5 = 105h+5=10

STEP 15

Subtract 5 from both sides of the equation to isolate the term with the variable.5h+5−5=10−55h + 5 - 5 = 10 - 55h+5−5=10−5

STEP 16

Simplify both sides of the equation.5h=55h = 55h=5

STEP 17

Divide both sides by 5 to solve for hhh.h=55h = \frac{5}{5}h=55​

STEP 18

Calculate the value of hhh.h=1h = 1h=1

STEP 19

Solve the fourth equation 60=−4(3y−6)60 = -4(3y - 6)60=−4(3y−6).Distribute the -4 on the right side of the equation.60=−4⋅3y+4⋅660 = -4 \cdot 3y + 4 \cdot 660=−4⋅3y+4⋅6

STEP 20

Simplify the right side of the equation.60=−12y+2460 = -12y + 2460=−12y+24

STEP 21

Subtract 24 from both sides of the equation to isolate the term with the variable.60−24=−12y60 - 24 = -12y60−24=−12y

STEP 22

Simplify both sides of the equation.36=−12y36 = -12y36=−12y

STEP 23

Divide both sides by -12 to solve for yyy.y=36−12y = \frac{36}{-12}y=−1236​

STEP 24

Calculate the value of yyy.y=−3y = -3y=−3

STEP 25

Solve the fifth equation −5g+24=7g−24-5g + 24 = 7g - 24−5g+24=7g−24.Add 5g5g5g to both sides of the equation to get all the ggg terms on one side.−5g+5g+24=7g+5g−24-5g + 5g + 24 = 7g + 5g - 24−5g+5g+24=7g+5g−24

STEP 26

Combine like terms.24=12g−2424 = 12g - 2424=12g−24

STEP 27

Add 24 to both sides of the equation to isolate the term with the variable.24+24=12g24 + 24 = 12g24+24=12g

STEP 28

Simplify both sides of the equation.48=12g48 = 12g48=12g

STEP 29

Divide both sides by 12 to solve for ggg.g=4812g = \frac{48}{12}g=1248​

STEP 30

Calculate the value of ggg.g=4g = 4g=4The solutions for the first five equations are:1. c=2c = 2c=22. x=11x = 11x=113. h=1h = 1h=14. y=−3y = -3y=−35. g=4g = 4g=4

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Assumptions
1. Each equation is a linear equation with one variable.
2. The equations are to be solved for the variable.
3. The solutions are real numbers.

Solve the first equation $7c - 3 + 2c = 15$ .
Combine like terms on the left side of the equation.
$7c + 2c - 3 = 15$

Add the coefficients of $c$ .
$9c - 3 = 15$

Add 3 to both sides of the equation to isolate the term with the variable.
$9c - 3 + 3 = 15 + 3$

Simplify both sides of the equation.
$9c = 18$

Divide both sides by 9 to solve for $c$ .
$c = \frac{18}{9}$

Calculate the value of $c$ .
$c = 2$

Solve the second equation $5(x - 7) = 20$ .
Distribute the 5 on the left side of the equation.
$5x - 35 = 20$

Add 35 to both sides of the equation to isolate the term with the variable.
$5x - 35 + 35 = 20 + 35$

Simplify both sides of the equation.
$5x = 55$

Divide both sides by 5 to solve for $x$ .
$x = \frac{55}{5}$

Calculate the value of $x$ .
$x = 11$

Solve the third equation $6h + 5 = 10 + h$ .
Subtract $h$ from both sides of the equation to get all the $h$ terms on one side.
$6h - h + 5 = 10$

Combine like terms.
$5h + 5 = 10$

Subtract 5 from both sides of the equation to isolate the term with the variable.
$5h + 5 - 5 = 10 - 5$

Simplify both sides of the equation.
$5h = 5$

Divide both sides by 5 to solve for $h$ .
$h = \frac{5}{5}$

Calculate the value of $h$ .
$h = 1$

Solve the fourth equation $60 = -4(3y - 6)$ .
Distribute the -4 on the right side of the equation.
$60 = -4 \cdot 3y + 4 \cdot 6$

Simplify the right side of the equation.
$60 = -12y + 24$

Subtract 24 from both sides of the equation to isolate the term with the variable.
$60 - 24 = -12y$

Simplify both sides of the equation.
$36 = -12y$

Divide both sides by -12 to solve for $y$ .
$y = \frac{36}{-12}$

Calculate the value of $y$ .
$y = -3$

Solve the fifth equation $-5g + 24 = 7g - 24$ .
Add $5g$ to both sides of the equation to get all the $g$ terms on one side.
$-5g + 5g + 24 = 7g + 5g - 24$

Combine like terms.
$24 = 12g - 24$

Add 24 to both sides of the equation to isolate the term with the variable.
$24 + 24 = 12g$

Simplify both sides of the equation.
$48 = 12g$

Divide both sides by 12 to solve for $g$ .
$g = \frac{48}{12}$

Calculate the value of $g$ .
$g = 4$
The solutions for the first five equations are:
1. $c = 2$
2. $x = 11$
3. $h = 1$
4. $y = -3$
5. $g = 4$