Question Solve linear equations with constant denominators. Examples: $\frac{x}{3}=\frac{x}{2}-2$ , $\frac{x}{5}=\frac{x}{6}+1$ , $20-\frac{x}{3}=\frac{x}{2}$ .

Studdy Solution

STEP 1

Assumptions
1. All equations are linear with variables in the numerator and constants in the denominator.
2. We need to solve for the variable $x$ in each equation.
3. The equations are independent of each other.
Let's solve each equation one by one.

STEP 2

Solve equation 17. $\frac{x}{3}=\frac{x}{2}-2$

STEP 3

Find a common denominator for the fractions on both sides of the equation. The common denominator for 3 and 2 is 6.

STEP 4

Multiply both sides of the equation by 6 to eliminate the denominators. $6 \cdot \frac{x}{3} = 6 \cdot \left(\frac{x}{2} - 2\right)$

STEP 5

Simplify both sides of the equation. $2x = 3x - 12$

STEP 6

Move all terms containing $x$ to one side and constants to the other side. $2x - 3x = -12$

STEP 7

Combine like terms. $-x = -12$

STEP 8

Divide both sides by -1 to solve for $x$ . $x = 12$

STEP 9

Solve equation 18. $\frac{x}{5}=\frac{x}{6}+1$

STEP 10

Find a common denominator for the fractions on both sides of the equation. The common denominator for 5 and 6 is 30.

STEP 11

Multiply both sides of the equation by 30 to eliminate the denominators. $30 \cdot \frac{x}{5} = 30 \cdot \left(\frac{x}{6} + 1\right)$

STEP 12

Simplify both sides of the equation. $6x = 5x + 30$

STEP 13

Move all terms containing $x$ to one side and constants to the other side. $6x - 5x = 30$

STEP 14

Combine like terms. $x = 30$

STEP 15

Solve equation 19. $20-\frac{x}{3}=\frac{x}{2}$

STEP 16

Find a common denominator for the fractions on both sides of the equation. The common denominator for 3 and 2 is 6.

STEP 17

Multiply both sides of the equation by 6 to eliminate the denominators. $6 \cdot 20 - 6 \cdot \frac{x}{3} = 6 \cdot \frac{x}{2}$

STEP 18

Simplify both sides of the equation. $120 - 2x = 3x$

STEP 19

Move all terms containing $x$ to one side and constants to the other side. $-2x - 3x = -120$

STEP 20

Combine like terms. $-5x = -120$

STEP 21

Divide both sides by -5 to solve for $x$ . $x = 24$

STEP 22

Solve equation 20. $\frac{x}{5}-\frac{1}{2}=\frac{x}{6}$

STEP 23

Find a common denominator for the fractions on both sides of the equation. The common denominator for 5, 2, and 6 is 30.

STEP 24

Multiply both sides of the equation by 30 to eliminate the denominators. $30 \cdot \left(\frac{x}{5} - \frac{1}{2}\right) = 30 \cdot \frac{x}{6}$

STEP 25

Simplify both sides of the equation. $6x - 15 = 5x$

STEP 26

Move all terms containing $x$ to one side and constants to the other side. $6x - 5x = 15$

STEP 27

Combine like terms. $x = 15$

STEP 28

Solve equation 21. $\frac{3 x}{5}=\frac{2 x}{3}+1$

STEP 29

Find a common denominator for the fractions on both sides of the equation. The common denominator for 5 and 3 is 15.

STEP 30

Multiply both sides of the equation by 15 to eliminate the denominators. $15 \cdot \frac{3 x}{5} = 15 \cdot \left(\frac{2 x}{3} + 1\right)$

STEP 31

Simplify both sides of the equation. $9x = 10x + 15$

STEP 32

Move all terms containing $x$ to one side and constants to the other side. $9x - 10x = -15$

STEP 33

Combine like terms. $-x = -15$

STEP 34

Divide both sides by -1 to solve for $x$ . $x = 15$

STEP 35

Solve equation 22. $\frac{x}{2}=\frac{3 x}{4}+5$

STEP 36

Find a common denominator for the fractions on both sides of the equation. The common denominator for 2 and 4 is 4.

STEP 37

Multiply both sides of the equation by 4 to eliminate the denominators. $4 \cdot \frac{x}{2} = 4 \cdot \left(\frac{3 x}{4} + 5\right)$

STEP 38

Simplify both sides of the equation. $2x = 3x + 20$

STEP 39

Move all terms containing $x$ to one side and constants to the other side. $2x - 3x = 20$

STEP 40

Combine like terms. $-x = 20$

STEP 41

Divide both sides by -1 to solve for $x$ . $x = -20$
The solutions are:
17. $x = 12$
18. $x = 30$
19. $x = 24$
20. $x = 15$
21. $x = 15$
22. $x = -20$

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