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Math  /  Algebra

Question−2(x+1)+5x=3(x+1)−2-2(x+1)+5 x=3(x+1)-2 No solution x=x= □\square All real•numbers are solutions 4(y−3)−6y=−2(y+6)4(y-3)-6 y=-2(y+6) No solution y=y= □\square All real numbers are solutions

Studdy Solution

STEP 1

1. We are given two separate equations to solve for x x and y y .
2. We need to determine whether each equation has no solution, a specific solution, or if all real numbers are solutions.
3. We will simplify each equation to check for contradictions or identities.

STEP 2

1. Solve the equation for x x .
2. Determine the solution type for the x x equation.
3. Solve the equation for y y .
4. Determine the solution type for the y y equation.

STEP 3

Solve the equation for x x :
−2(x+1)+5x=3(x+1)−2 -2(x+1) + 5x = 3(x+1) - 2
First, distribute the terms:
−2x−2+5x=3x+3−2 -2x - 2 + 5x = 3x + 3 - 2
Combine like terms:
(5x−2x)−2=3x+1 (5x - 2x) - 2 = 3x + 1
3x−2=3x+1 3x - 2 = 3x + 1

STEP 4

Determine the solution type for the x x equation:
Subtract 3x 3x from both sides:
3x−2−3x=3x+1−3x 3x - 2 - 3x = 3x + 1 - 3x
−2=1 -2 = 1
This is a contradiction, meaning there is no solution for x x .

STEP 5

Solve the equation for y y :
4(y−3)−6y=−2(y+6) 4(y-3) - 6y = -2(y+6)
Distribute the terms:
4y−12−6y=−2y−12 4y - 12 - 6y = -2y - 12
Combine like terms:
(4y−6y)−12=−2y−12 (4y - 6y) - 12 = -2y - 12
−2y−12=−2y−12 -2y - 12 = -2y - 12

STEP 6

Determine the solution type for the y y equation:
Add 2y 2y to both sides:
−2y+2y−12=−2y+2y−12 -2y + 2y - 12 = -2y + 2y - 12
−12=−12 -12 = -12
This is an identity, meaning all real numbers are solutions for y y .
The solution for the x x equation is: No solution. The solution for the y y equation is: All real numbers are solutions.

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