Question Video: Using the Slope of a Straight Line to Solve a Problem

Video Transcript

If the slope of the straight line three 𝑎 plus seven all multiplied by 𝑥 plus four 𝑎𝑦 plus four is equal to zero equals negative one, find the value of 𝑎.

In this question, we’re given the equation of a straight line, and we’re told that the slope of this straight line is equal to negative one. We need to use this information to determine the value of 𝑎. To answer this question, we need to notice some things: first, the form we’re given the equation of our straight line in. This is called the general form of the equation of a straight line, and it’s not easy to find the slope from this form.

However, we do know a form of straight line where it’s easy to find the slope of our line. We know the slope–intercept form of a straight line is the equation 𝑦 is equal to 𝑚𝑥 plus 𝑏, where 𝑚 is the slope of our straight line and 𝑏 is the 𝑦-intercept of our straight line. If we could rearrange the equation for the straight line given to us into this form, then we know that the coefficient of 𝑥 must be the slope. It must be equal to negative one. And in fact, we can almost always do this. The only time this will be complicated is if we’re dealing with a vertical line because when we’re dealing with a vertical line, we won’t have a value for the slope.

However, in this case, we know the slope is equal to negative one, so we don’t need to worry about these two cases. This means we know we can always find the equation in this form. To do this, we want 𝑦 to be on its own on the left-hand side of our equation. So let’s start by subtracting both terms which don’t involve 𝑦. So we subtract three 𝑎 plus seven times 𝑥 from both sides of our equation, and we subtract four from both sides of our equation. This gives us the equation four 𝑎𝑦 is equal to negative one times three 𝑎 plus seven multiplied by 𝑥 minus four.

Next, in our slope–intercept form, the coefficient of 𝑦 needs to be one. So we’re going to need to divide through both sides of our equation through by four 𝑎. And it’s worth reiterating here we know the value of 𝑎 is not equal to zero because if our value of 𝑎 was equal to zero, we would lose the 𝑦-term in our general equation for the straight line. Then, if we were to solve this equation, we will get one solution for 𝑥 and we could have any value for 𝑦. In other words, we get a vertical line. But we know that we don’t have a vertical line because we know the slope is equal to negative one. So we divide both sides of our equations through by four 𝑎. We get 𝑦 is equal to negative one times three 𝑎 plus seven multiplied by 𝑥 minus four all divided by four 𝑎.

But we want to write this in slope–intercept form. So we’re going to split this in our numerator. This gives us that 𝑦 is equal to negative one times three 𝑎 plus seven all over four 𝑎 all multiplied by 𝑥 minus four divided by four 𝑎. And we can simplify this although it’s not necessary; in our second term, four divided by four is equal to one. Remember, in the slope–intercept form of a straight line, the coefficient of 𝑥 is the slope of our line, and we’re told in the question this is equal to negative one. Therefore, we can get an equation for 𝑎 if we just set the coefficient of 𝑥 equal to negative one; we get the equation negative one is equal to negative one times three 𝑎 plus seven all divided by four 𝑎.

Now all we need to do is solve this equation for 𝑎. We’ll start by multiplying through by four 𝑎. We get negative four 𝑎 is equal to negative one times three 𝑎 plus seven. Next, we’re going to distribute the negative over our parentheses. This gives us negative four 𝑎 is equal to negative three 𝑎 minus seven. Next, we’re going to add three 𝑎 to both sides of our equation. This gives us negative four 𝑎 plus three 𝑎 is equal to negative seven. We know that negative four plus three is equal to negative one. So this equation simplifies to give us that negative 𝑎 is equal to negative seven and we can solve for our value of 𝑎 by multiplying through by negative one. We get that 𝑎 is equal to seven, which is our final answer.

Therefore, given that the slope of the straight line three 𝑎 plus seven all multiplied by 𝑥 plus four 𝑎𝑦 plus four is equal to zero was equal to negative one, then, by writing this equation in slope–intercept form, we were able to show that the value of 𝑎 had to be equal to seven.