A technician uses a linear model to estimate the time, t hours, required to complete a pro… — ACT

A technician uses a linear model to estimate the time, $t$ hours, required to complete a project. The estimate satisfies the equation:

\frac{2}{3}(t - 12) + \frac{5}{6}(t + 6) = 18

What is the value of $t$ ?

Answer: 14 hours

The value of

t

14

To solve the linear equation, we first multiply all terms by the least common multiple of the denominators, which is

6

. This simplifies the equation to

4(t - 12) + 5(t + 6) = 108

. Distributing the terms gives

4t - 48 + 5t + 30 = 108

. Combining like terms yields

9t - 18 = 108

. Adding

18

to both sides gives

9t = 126

. Finally, dividing by

9

gives

t = 14

Multiply both sides of the equation by

6

to eliminate the denominators.

4(t - 12) + 5(t + 6) = 108

Multiplying by the least common multiple of the denominators clears the fractions, making the linear equation easier to solve.

Apply the distributive property to expand the terms.

4t - 48 + 5t + 30 = 108

Distributing the constants allows us to group like terms.

Combine the variable terms and the constant terms on the left side.

9t - 18 = 108

Simplifying the expression is a necessary step before isolating the variable.

Add

18

to both sides of the equation.

9t = 126

Adding the constant to both sides isolates the variable term on the left side.

Divide both sides of the equation by

9

to solve for

t

t = 14

Dividing by the coefficient of the variable yields the final solution.

Solving linear equations with fractional coefficients by clearing denominators and isolating the variable.

Question