Question Video: Completing a Table of Values for a Linear Relation and Drawing Its Graph Mathematics

Video Transcript

Given the linear relation negative four 𝑥 plus two 𝑦 is equal to negative six, complete the table of values below. Using the complete table, sketch a graph to represent the relation.

In this example, we first need to fill in the missing values in the table, where each ordered pair 𝑥 and 𝑦 must satisfy the given linear relation. We can then use the values from the completed table to sketch a graph of the relation.

So let’s begin by reminding ourselves of what we mean by a linear relation. A relation of the form 𝑎𝑥 plus 𝑏𝑦 equals 𝑐 is a linear relation that can be represented by a set of ordered pairs 𝑥, 𝑦, where each pair of values satisfies the given equation for specific constants 𝑎, 𝑏, and 𝑐. For four out of the five ordered pairs, 𝑥, 𝑦, in the table, we’re given either an 𝑥-value or a 𝑦-value. And so we must solve the equation, negative four 𝑥 plus two 𝑦 equals negative six, to find the missing value in each of these four ordered pairs.

So let’s begin with the first ordered pair, where we’re given that 𝑦 equals negative seven. And we must find the 𝑥-value that together with this 𝑦-value satisfies the given equation. To do this, we substitute 𝑦 equals negative seven into the equation, which gives us negative four 𝑥 minus 14 is equal to negative six. Now, adding 14 to both sides, we have negative four 𝑥 is equal to eight. And dividing both sides by negative four, we isolate 𝑥 on the left-hand side. And we’re left with negative two on the right. So, if 𝑦 is negative seven, then the value of 𝑥 satisfying the given linear relation must be negative two. And we can put this value in the table. Although we’re not specifically asked to, we can write the ordered pair negative two, negative seven as shown below the table.

In the second ordered pair, we’re given the 𝑥-value, that is, 𝑥 equal to zero. So, this time, we have to solve for 𝑦. Substituting 𝑥 equal to zero into the equation for the linear relation, we have negative four times zero plus two 𝑦 equals negative six. Since negative four times zero is equal to zero, this leaves us with two 𝑦 equals negative six. We can isolate 𝑦 on the left by dividing both sides by two. And so we’re left with 𝑦 equals negative three. Hence, when 𝑥 is zero in the relation negative four 𝑥 plus two 𝑦 equals negative six, 𝑦 must be equal to negative three. And we can put this value into the table with the ordered pair zero, negative three below.

In the next column of the table, we’re given both values: 𝑥 equals one and 𝑦 equals negative one. So the ordered pair is one, negative one. Now, moving on to the fourth column, where we’re given that 𝑦 equals positive three, with this value substituted into the equation, we have negative four 𝑥 plus six equals negative six. Subtracting six from both sides and dividing through by negative four, we find when 𝑦 equals positive three, 𝑥 is also positive three.

Finally, in the last column, we’re given that 𝑥 is equal to five. Substituting this into the equation, we have negative 20 plus two 𝑦 equals negative six. Adding 20 to both sides and dividing through by two, we find 𝑦 is equal to seven. And the ordered pair is then five, seven. Hence, given the linear relation negative four 𝑥 plus two 𝑦 equals negative six, we find the missing values are negative two, negative three, positive three, and seven.

Now, for the second part of this question, using the values from the completed table, we’re going to sketch a graph to represent the given relation. Note that we’ve already written out the ordered pairs that are the coordinates of points on the line representing this relation. So all we need to do is plot the points with these coordinates on our graph and draw a line through them. The first point has coordinates 𝑥 is negative two and 𝑦 is negative seven, which we can plot on our graph as shown. The second has coordinates zero, negative three; the third, one, negative one; and the fourth and fifth points have coordinates three, three and five, seven, respectively.

Our final step is to draw a line through these five points. And with this line, we’ve sketched a graph that represents the relation negative four 𝑥 plus two 𝑦 equals negative six.