Question Video: Finding the Value of an Unknown Coefficient in a Linear Equation given an Ordered Pair That Satisfies It Mathematics • Third Year of Preparatory School

Video Transcript

Given that the ordered pair negative seven, negative three satisfies the relation 𝑓 of 𝑥 equals negative three plus three 𝑥 over 𝑏, find the value of 𝑏.

Let’s recall what it means for an ordered pair to satisfy some relation. The general form of an ordered pair that will satisfy some relation for a function 𝑓 of 𝑥 is 𝑥 and 𝑓 of 𝑥. In other words, for the ordered pair negative seven, negative three, if 𝑥 equals negative seven, this tells us that the corresponding 𝑓 of 𝑥 value, 𝑓 of negative seven, must be equal to negative three. And so we can substitute these two values into our equation for the relation. 𝑓 of 𝑥 is negative three plus three 𝑥 over 𝑏. 𝑓 of negative seven will be negative three plus three times negative seven over 𝑏. We replace 𝑥 with negative seven. This simplifies to negative three minus 21 over 𝑏, which is negative negative 18 over 𝑏.

Since a negative divided by a negative is negative and then a negative times a negative is positive, we can rewrite this as 18 over 𝑏. But of course we know that 𝑓 of negative seven is also equal to negative three. So our equation becomes negative three equals 18 over 𝑏. We’re going to solve for 𝑏 by first multiplying both sides of the equation by 𝑏, giving us negative three 𝑏 equals 18. Finally, we can divide through by negative three. 18 divided by negative three is negative six. So we found the value of 𝑏 to be equal to negative six. Given that our ordered pair satisfies the relation for 𝑓 of 𝑥 then, 𝑏 must be equal to negative six.