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

# Question Video: Using the Slope of a Straight Line to Solve a Problem Mathematics • First Year of Secondary School

## Join Nagwa Classes

If the slope of the straight line (3๐ + 7)๐ฅ + 4๐๐ฆ + 4 = 0 equals โ1, find the value of ๐.

03:37

### Video Transcript

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

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. 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๐ฅ plus ๐, where ๐ 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 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๐ฆ is equal to negative one times three ๐ 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. And itโs worth reiterating here we know the value of ๐ is not equal to zero because if our value of ๐ 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. We get ๐ฆ is equal to negative one times three ๐ plus seven multiplied by ๐ฅ minus four all divided by four ๐.

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 plus seven all over four ๐ all multiplied by ๐ฅ minus four divided by four ๐. 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 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 plus seven all divided by four ๐.

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

Therefore, given that the slope of the straight line three ๐ plus seven all multiplied by ๐ฅ plus four ๐๐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 had to be equal to seven.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions