Question15. What is the solution to $4(y-3)+19=8(2 y+3)+7$ ?

Studdy Solution

STEP 1

1. The equation $4(y-3) + 19 = 8(2y+3) + 7$ is a linear equation in one variable, $y$ .
2. The goal is to find the value of $y$ that satisfies the equation.
3. We will use basic algebraic operations such as distribution, combining like terms, and isolating the variable.

STEP 2

1. Simplify both sides of the equation by distributing and combining like terms.
2. Move all terms involving $y$ to one side of the equation and constant terms to the other side.
3. Solve for $y$ by isolating the variable.
4. Check the solution by substituting it back into the original equation.

STEP 3

Distribute the constants on both sides of the equation:
Left side: $4(y - 3) = 4y - 12$
Right side: $8(2y + 3) = 16y + 24$
The equation becomes: $4y - 12 + 19 = 16y + 24 + 7$

STEP 4

Combine like terms on both sides:
Left side: $4y + 7$
Right side: $16y + 31$
The equation now is: $4y + 7 = 16y + 31$

STEP 5

Move all terms involving $y$ to one side and constant terms to the other side. Subtract $4y$ from both sides:
$7 = 12y + 31$
Subtract 31 from both sides to isolate terms with $y$ :
$7 - 31 = 12y$

STEP 6

Simplify the equation:
$-24 = 12y$

STEP 7

Solve for $y$ by dividing both sides by 12:
$y = \frac{-24}{12}$
Simplify:
$y = -2$

STEP 8

Check the solution by substituting $y = -2$ back into the original equation:
Original equation: $4(y-3) + 19 = 8(2y+3) + 7$
Substitute $y = -2$ :
Left side: $4(-2 - 3) + 19 = 4(-5) + 19 = -20 + 19 = -1$
Right side: $8(2(-2) + 3) + 7 = 8(-4 + 3) + 7 = 8(-1) + 7 = -8 + 7 = -1$
Both sides are equal, confirming the solution is correct.
The solution is: $\boxed{-2}$

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