Math

Question Solve the equations 3(x4)=x+23(x-4)=x+2, 23x12=13x+52\frac{2}{3}x-\frac{1}{2}=\frac{1}{3}x+\frac{5}{2}, and 15(10x5)=3x+2\frac{1}{5}(10x-5)=3x+2. Determine which statements are true.

Studdy Solution

STEP 1

Assumptions
1. We are given three separate equations to solve: a. 3(x4)=x+23(x - 4) = x + 2 b. 23x12=13x+52\frac{2}{3} x - \frac{1}{2} = \frac{1}{3} x + \frac{5}{2} c. 15(10x5)=3x+2\frac{1}{5}(10x - 5) = 3x + 2
2. We will solve each equation for xx.
3. We are looking for the truth value of certain statements after solving the equations.

STEP 2

Solve the first equation 3(x4)=x+23(x - 4) = x + 2.

STEP 3

Distribute the 33 on the left side of the equation.
3x12=x+23x - 12 = x + 2

STEP 4

Subtract xx from both sides of the equation to move all xx terms to one side.
3xx12=xx+23x - x - 12 = x - x + 2

STEP 5

Simplify both sides of the equation.
2x12=22x - 12 = 2

STEP 6

Add 1212 to both sides of the equation to isolate the term with xx.
2x12+12=2+122x - 12 + 12 = 2 + 12

STEP 7

Simplify both sides of the equation.
2x=142x = 14

STEP 8

Divide both sides of the equation by 22 to solve for xx.
x=142x = \frac{14}{2}

STEP 9

Calculate the value of xx.
x=7x = 7

STEP 10

Solve the second equation 23x12=13x+52\frac{2}{3} x - \frac{1}{2} = \frac{1}{3} x + \frac{5}{2}.

STEP 11

To eliminate fractions, find a common denominator for all the terms, which is 66 in this case, and multiply each term by 66.
6(23x12)=6(13x+52)6 \left(\frac{2}{3} x - \frac{1}{2}\right) = 6 \left(\frac{1}{3} x + \frac{5}{2}\right)

STEP 12

Distribute the 66 and simplify each term.
4x3=2x+154x - 3 = 2x + 15

STEP 13

Subtract 2x2x from both sides of the equation to move all xx terms to one side.
4x2x3=2x2x+154x - 2x - 3 = 2x - 2x + 15

STEP 14

Simplify both sides of the equation.
2x3=152x - 3 = 15

STEP 15

Add 33 to both sides of the equation to isolate the term with xx.
2x3+3=15+32x - 3 + 3 = 15 + 3

STEP 16

Simplify both sides of the equation.
2x=182x = 18

STEP 17

Divide both sides of the equation by 22 to solve for xx.
x=182x = \frac{18}{2}

STEP 18

Calculate the value of xx.
x=9x = 9

STEP 19

Solve the third equation 15(10x5)=3x+2\frac{1}{5}(10x - 5) = 3x + 2.

STEP 20

Distribute the 15\frac{1}{5} on the left side of the equation.
2x1=3x+22x - 1 = 3x + 2

STEP 21

Subtract 2x2x from both sides of the equation to move all xx terms to one side.
2x2x1=3x2x+22x - 2x - 1 = 3x - 2x + 2

STEP 22

Simplify both sides of the equation.
1=x+2-1 = x + 2

STEP 23

Subtract 22 from both sides of the equation to solve for xx.
12=x+22-1 - 2 = x + 2 - 2

STEP 24

Calculate the value of xx.
x=3x = -3

STEP 25

Now that we have solved all three equations, we have the solutions:
1. x=7x = 7 for the first equation.
2. x=9x = 9 for the second equation.
3. x=3x = -3 for the third equation.

We can now evaluate the truth value of the statements based on these solutions.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord