Question 126·Medium·Algebra

After distributing, simplify each side by combining the constant numbers together and rewriting the equation in the form "(coefficient)$\cdot k$ + constant = (coefficient)$\cdot k$ + constant."

Subtract one of the $k$-terms from both sides so that all the $k$'s are on one side and all the constants are on the other, then solve the resulting simple equation.

When you subtract $(\tfrac12)k$ from $(\tfrac34)k$, rewrite the fractions with a common denominator before subtracting the numerators.

In Desmos, type y1 = (3/4)(x - 8) + 5 on one line and y2 = (1/2)(x + 4) + 9 on the next line. These represent the left and right sides of the equation as functions of $x$.

Look for the point where the two graphs intersect. You can tap/click the intersection point; Desmos will display its coordinates $(x, y)$.

The $x$-coordinate of the intersection is the value of $k$ that makes both sides of the original equation equal. Match this $x$-value to the correct answer choice.

Apply the distributive property to remove the parentheses.

• Left side:

  • $(\tfrac34)(k - 8) = (\tfrac34)k - (\tfrac34)\cdot 8$
  • $(\tfrac34)\cdot 8 = \dfrac{3\cdot 8}{4} = \dfrac{24}{4} = 6$
  • So the left side becomes $(\tfrac34)k - 6 + 5$, which simplifies to $(\tfrac34)k - 1$.

• Right side:

  • $(\tfrac12)(k + 4) = (\tfrac12)k + (\tfrac12)\cdot 4$
  • $(\tfrac12)\cdot 4 = \dfrac{4}{2} = 2$
  • So the right side becomes $(\tfrac12)k + 2 + 9$, which simplifies to $(\tfrac12)k + 11$.

Now the equation is:

Subtract $(\tfrac12)k$ from both sides to keep all $k$ terms together:

Compute the coefficient of $k$ by subtracting the fractions:

So the equation becomes:

First, isolate the term with $k$ by adding 1 to both sides:

Now, multiply both sides by 4 to solve for $k$:

So the value of $k$ that satisfies the equation is $48$, which corresponds to answer choice D.