Question

Solve the following equation \[13\left( {y - 4} \right) - 3\left( {y - 9} \right) - 5\left( {y + 4} \right) = 0\]

Answer 1

Hint: In this question we have to solve the equation and get the value of y .First, we simplify the terms involving brackets by letting multiplying factors into it. Next, we gather the like terms and combine the like terms in the equation. Then we combine the constant terms. Later we simplify the remaining equation and get the value of y. Now after getting the value of y we verify the value of y by equating left hand side and right hand side.

Complete step-by-step solution:
The given equation is \[13\left( {y - 4} \right) - 3\left( {y - 9} \right) - 5\left( {y + 4} \right) = 0\].
\[
13\left( {y - 4} \right) - 3\left( {y - 9} \right) - 5\left( {y + 4} \right) = 0 \\
\Rightarrow 13y - 4 \times 13 - 3\left( {y - 9} \right) - 5\left( {y + 4} \right) = 0 \\
\Rightarrow 13y - 52 - 3\left( {y - 9} \right) - 5\left( {y + 4} \right) = 0 \\
\Rightarrow 13y - 52 - 3y - ( - 3) \times 9 - 5\left( {y + 4} \right) = 0 \\
\Rightarrow 13y - 52 - 3y + 27 - 5\left( {y + 4} \right) = 0 \\
\Rightarrow 13y - 52 - 3y + 27 - 5y + ( - 5) \times 4 = 0 \\
\Rightarrow 13y - 52 - 3y + 27 - 5y - 20 = 0 \\
\Rightarrow 13y - 3y - 5y - 52 + 27 - 20 = 0 \\
\Rightarrow (13 - 3 - 5)y - 52 + 27 - 20 = 0 \\
\Rightarrow (10 - 5)y - 52 + 27 - 20 = 0 \\
\Rightarrow 5y - 52 + 27 - 20 = 0 \\
\Rightarrow 5y - 52 + 7 = 0 \\
\Rightarrow 5y - 45 = 0 \
\Rightarrow 5y = 45 \\
\Rightarrow y = 9 \]
Hence we got the value of y =9.
Verification: Now substitute the value of \[y = 9\] in equation \[13\left( {y - 4} \right) - 3\left( {y - 9} \right) - 5\left( {y + 4} \right) = 0\] and whether left hand side is equal to right hand side (L .H. S =R .H. S).
\[ 13\left( {y - 4} \right) - 3\left( {y - 9} \right) - 5\left( {y + 4} \right) = 0 \\
\Rightarrow 13(9 - 4) - 3(9 - 9) - 5(9 + 4) = 0 \\
\Rightarrow 13(9 - 4) - 5(9 + 4) = 0 \\
\Rightarrow 13 \times 5 - 5 \times 13 = 0 \\
\Rightarrow 0 = 0 \]
L .H. S = R.H. S
Hence it is verified that the value of y for the the equation \[13\left( {y - 4} \right) - 3\left( {y - 9} \right) - 5\left( {y + 4} \right) = 0\] is 9.

Note: The above given equation \[13\left( {y - 4} \right) - 3\left( {y - 9} \right) - 5\left( {y + 4} \right) = 0\] is a linear equation in single variable. Solving the equation means getting the value of y. For solving equations like above we have to first simplify terms associated with parenthesis and brackets and then move forward for the variable associated terms. Lastly, we combine constants and then we get the value of variables.
Checking the terms of left hand side and right hand side assures us with a perfect answer. As we are solving algebraic expressions and equations, verification (L. H .S = R. H. S) is needed for getting correct answers.