Question Video: Evaluating Algebraic Expressions Involving Factorization by Taking Out the Highest Common Factor

Video Transcript

If five 𝑑 minus three 𝑓 is equal to negative three and two 𝑝 plus three 𝑑 is equal to negative four, what is the value of 12𝑑 plus six 𝑓 minus 10𝑑 plus eight 𝑝?

The sum of the four terms 12𝑑, six 𝑓, negative 10𝑑, and eight 𝑝 can be written in any order. This means that 12𝑑 plus six 𝑓 minus 10𝑑 plus eight 𝑝 is equal to negative 10𝑑 plus six 𝑓 plus eight 𝑝 plus 12𝑑.

Let’s begin by considering one of the equations in the question, five 𝑑 minus three 𝑓 is equal to negative three. We can multiply both sides of this equation by negative two. This gives us negative 10𝑑 plus six 𝑓 is equal to six. We recall that when multiplying two negative terms, we get a positive answer. The left-hand side here corresponds to the first two terms of our new expression. Negative 10𝑑 plus six 𝑓 is equal to six.

Let’s now consider the second equation we are given, two 𝑝 plus three 𝑑 is equal to negative four. We can multiply both sides of this equation by four. On the left-hand side, we have eight 𝑝 plus 12𝑑. On the right-hand side, we have negative 16. This means that we can replace eight 𝑝 plus 12𝑑 in our expression with negative 16. Adding negative 16 to six is the same as subtracting 16 from six. This is equal to negative 10.

If five 𝑑 minus three 𝑓 is equal to negative three and two 𝑝 plus three 𝑑 is negative four, then 12𝑑 plus six 𝑓 minus 10𝑑 plus eight 𝑝 is equal to negative 10.