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

If the slope of the straight line (3๐‘Ž + 7)๐‘ฅ + 4๐‘Ž๐‘ฆ + 4 = 0 equals โˆ’1, find the value of ๐‘Ž.

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Video Transcript

If the slope of the straight line three ๐‘Ž plus seven all multiplied by ๐‘ฅ plus four ๐‘Ž๐‘ฆ plus four is equal to zero equals negative one, find the value of ๐‘Ž.

In this question, weโ€™re given the equation of a straight line, and weโ€™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 ๐‘Ž. To answer this question, we need to notice some things: first, the form weโ€™re given the equation of our straight line in. This is called the general form of the equation of a straight line, and itโ€™s not easy to find the slope from this form.

However, we do know a form of straight line where itโ€™s easy to find the slope of our line. We know the slopeโ€“intercept form of a straight line is the equation ๐‘ฆ is equal to ๐‘š๐‘ฅ plus ๐‘, where ๐‘š is the slope of our straight line and ๐‘ is the ๐‘ฆ-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 ๐‘ฅ 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 weโ€™re dealing with a vertical line because when weโ€™re dealing with a vertical line, we wonโ€™t have a value for the slope.

However, in this case, we know the slope is equal to negative one, so we donโ€™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 ๐‘ฆ to be on its own on the left-hand side of our equation. So letโ€™s start by subtracting both terms which donโ€™t involve ๐‘ฆ. So we subtract three ๐‘Ž plus seven times ๐‘ฅ from both sides of our equation, and we subtract four from both sides of our equation. This gives us the equation four ๐‘Ž๐‘ฆ is equal to negative one times three ๐‘Ž plus seven multiplied by ๐‘ฅ minus four.

Next, in our slopeโ€“intercept form, the coefficient of ๐‘ฆ needs to be one. So weโ€™re going to need to divide through both sides of our equation through by four ๐‘Ž. And itโ€™s worth reiterating here we know the value of ๐‘Ž is not equal to zero because if our value of ๐‘Ž was equal to zero, we would lose the ๐‘ฆ-term in our general equation for the straight line. Then, if we were to solve this equation, we will get one solution for ๐‘ฅ and we could have any value for ๐‘ฆ. In other words, we get a vertical line. But we know that we donโ€™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 ๐‘Ž. We get ๐‘ฆ is equal to negative one times three ๐‘Ž plus seven multiplied by ๐‘ฅ minus four all divided by four ๐‘Ž.

But we want to write this in slopeโ€“intercept form. So weโ€™re going to split this in our numerator. This gives us that ๐‘ฆ is equal to negative one times three ๐‘Ž plus seven all over four ๐‘Ž all multiplied by ๐‘ฅ minus four divided by four ๐‘Ž. And we can simplify this although itโ€™s not necessary; in our second term, four divided by four is equal to one. Remember, in the slopeโ€“intercept form of a straight line, the coefficient of ๐‘ฅ is the slope of our line, and weโ€™re told in the question this is equal to negative one. Therefore, we can get an equation for ๐‘Ž if we just set the coefficient of ๐‘ฅ equal to negative one; we get the equation negative one is equal to negative one times three ๐‘Ž plus seven all divided by four ๐‘Ž.

Now all we need to do is solve this equation for ๐‘Ž. Weโ€™ll start by multiplying through by four ๐‘Ž. We get negative four ๐‘Ž is equal to negative one times three ๐‘Ž plus seven. Next, weโ€™re going to distribute the negative over our parentheses. This gives us negative four ๐‘Ž is equal to negative three ๐‘Ž minus seven. Next, weโ€™re going to add three ๐‘Ž to both sides of our equation. This gives us negative four ๐‘Ž plus three ๐‘Ž 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 ๐‘Ž is equal to negative seven and we can solve for our value of ๐‘Ž by multiplying through by negative one. We get that ๐‘Ž is equal to seven, which is our final answer.

Therefore, given that the slope of the straight line three ๐‘Ž plus seven all multiplied by ๐‘ฅ plus four ๐‘Ž๐‘ฆ plus four is equal to zero was equal to negative one, then, by writing this equation in slopeโ€“intercept form, we were able to show that the value of ๐‘Ž had to be equal to seven.

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