QuestionSolve the equation and check. $y=\square$

Studdy Solution

STEP 1

1. The equation is a linear equation in the variable $y$ .
2. The goal is to solve for $y$ by isolating it on one side of the equation.
3. Simplification will be done using basic algebraic operations such as distribution, combining like terms, and solving linear equations.

STEP 2

1. Distribute the constants into the parentheses on both sides.
2. Combine like terms on both sides.
3. Isolate $y$ on one side of the equation.
4. Solve for $y$ and check the solution by substituting it back into the original equation.

STEP 3

Distribute the constants into the parentheses on both sides of the equation.
$4(5y - 2) + 3(2y + 6) = 25(3y + 2) - 14y$
$4 \cdot 5y + 4 \cdot (-2) + 3 \cdot 2y + 3 \cdot 6 = 25 \cdot 3y + 25 \cdot 2 - 14y$
$20y - 8 + 6y + 18 = 75y + 50 - 14y$

STEP 4

Combine like terms on both sides of the equation.
$(20y + 6y) + (-8 + 18) = (75y - 14y) + 50$
$26y + 10 = 61y + 50$

STEP 5

Isolate $y$ on one side of the equation by first subtracting $26y$ from both sides.
$26y + 10 - 26y = 61y + 50 - 26y$
$10 = 35y + 50$
Next, subtract 50 from both sides to further isolate the term with $y$ .
$10 - 50 = 35y + 50 - 50$
$-40 = 35y$

STEP 6

Solve for $y$ by dividing both sides by 35.
$y = \frac{-40}{35}$
$y = \frac{-8}{7}$

STEP 7

Check the solution by substituting $y = \frac{-8}{7}$ back into the original equation.
$4(5y - 2) + 3(2y + 6) = 25(3y + 2) - 14y$
Substitute $y = \frac{-8}{7}$ :
$4\left(5\left(\frac{-8}{7}\right) - 2\right) + 3\left(2\left(\frac{-8}{7}\right) + 6\right) = 25\left(3\left(\frac{-8}{7}\right) + 2\right) - 14\left(\frac{-8}{7}\right)$
Simplify each term:
$4\left(\frac{-40}{7} - 2\right) + 3\left(\frac{-16}{7} + 6\right) = 25\left(\frac{-24}{7} + 2\right) + \frac{112}{7}$
Convert constants to have a common denominator:
$4\left(\frac{-40 - 14}{7}\right) + 3\left(\frac{-16 + 42}{7}\right) = 25\left(\frac{-24 + 14}{7}\right) + \frac{112}{7}$
$4\left(\frac{-54}{7}\right) + 3\left(\frac{26}{7}\right) = 25\left(\frac{-10}{7}\right) + \frac{112}{7}$
$\frac{-216}{7} + \frac{78}{7} = \frac{-250}{7} + \frac{112}{7}$
$\frac{-138}{7} = \frac{-138}{7}$
The solution is verified.
The solution to the equation is:
$y = \frac{-8}{7}$

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