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2.5 O) Gradient with Unknowns
2.5 O) Gradient with Unknowns
The content in this section builds on the content that was discussed in sections F and G. Therefore, make sure that you have covered all of the content in these two sections before covering the content in this section (click here to be taken to section F, and click here to be taken to section G).
It may be the case in the exam that you are given the coordinates of two points and one of the values in the two coordinates is an unknown (like a or b). You may then be given the gradient of the line that connects the two points and asked to find the value of the missing coordinate. We answer questions like this by subbing in the values from the two points into the formula for working out the gradient of a line from two points. We then solve to find the value of the unknown. The formula for working out the gradient of a line from two points is given below.
It may be the case in the exam that you are given the coordinates of two points and one of the values in the two coordinates is an unknown (like a or b). You may then be given the gradient of the line that connects the two points and asked to find the value of the missing coordinate. We answer questions like this by subbing in the values from the two points into the formula for working out the gradient of a line from two points. We then solve to find the value of the unknown. The formula for working out the gradient of a line from two points is given below.
There are two examples in this section. The first example is where the unknown value is a y coordinate and the second example is where the unknown value is a x coordinate.
Example 1
There are two points; point A and point B. Point A has the coordinates (1, 5) and point B has the coordinates (3, c).
The gradient of the line that connects points A and B is 4. Work out the value of c.
We are going to answer this question by using the formula for working out the gradient of a line from two points. We sub in all of the values that we are given into their appropriate places in the formula, and then solve to find the value for the unknown that we are looking for. The formula for working out the gradient from two points is given below:
There are two points; point A and point B. Point A has the coordinates (1, 5) and point B has the coordinates (3, c).
The gradient of the line that connects points A and B is 4. Work out the value of c.
We are going to answer this question by using the formula for working out the gradient of a line from two points. We sub in all of the values that we are given into their appropriate places in the formula, and then solve to find the value for the unknown that we are looking for. The formula for working out the gradient from two points is given below:
We now need to label our two points point 1 and point 2. I am going to have A as point 1 and B as point 2. We now label up the x and y coordinates for these points and the labelled up coordinates are shown below:
We were told in the question that the gradient of the line that connects the two points is 4. We now sub the values from the two points (x1, y1, x2 and y2) and the gradient into the formula.
We can tidy up the denominator of the fraction on the left side of the equation.
We now need to solve the equation to work out the value of c. The first step in finding the value of c is to get rid of the fraction on the left side of the equation. We are able to get rid of the fraction by multiplying both sides of the equation by the denominator of the fraction. Therefore, we multiply both sides of the equation by 2.
The next step in finding the value of c is to move the -5 that is currently on the left side of the equation to the right. We are able to do this by doing the opposite. The opposite of taking 5 is adding 5, so we add 5 to both sides of the equation.
Therefore, c is 13.
Example 2
In the previous example, we had an unknown y coordinate. We are now going to have an example whereby we have an unknown x coordinate. We answer questions like this in pretty much the same way.
There are two points; point A and point B. Point A has the coordinates (-4, -6) and point B has the coordinates (d, 3).
The gradient of the line that connects point A and point B is -3. Work out the value of d.
The first step in answering this question is to write down the formula for working out the gradient of a line from two points. The formula is given below:
In the previous example, we had an unknown y coordinate. We are now going to have an example whereby we have an unknown x coordinate. We answer questions like this in pretty much the same way.
There are two points; point A and point B. Point A has the coordinates (-4, -6) and point B has the coordinates (d, 3).
The gradient of the line that connects point A and point B is -3. Work out the value of d.
The first step in answering this question is to write down the formula for working out the gradient of a line from two points. The formula is given below:
The next step is to label the points as point 1 and point 2. I am going to have A as point 1 and B as point 2. We now label up the x and y coordinates for these points, and the labelled up coordinates are shown below:
We were told in the question that the gradient of the line that connects the two points is -3. We now sub the values from the two points (x1, y1, x2 and y2) and the gradient into the formula. It is best to put any negative values that are being subtracted in brackets, and this is so that we do not make any mistakes with the signs.
The taking of a negative number results in the number being added. Therefore, the –(-6) on the numerator becomes +6, and the –(-4) on the denominator becomes +4.
The numerator can be simplified to become 9.
We now need to solve the equation to find the value of d. The first step in finding the value of d is to get d off of the denominator of the fraction. We are able to do this by multiplying both sides of the equation by the denominator of the fraction; we multiply both sides of the equation by “d + 4”. When we are multiplying the right side of the equation by “d + 4”, it is best to have “d + 4” in a bracket.
We now expand out the bracket on the right side of the equation. We do this by multiplying the term that is on the outside of the bracket (-3) by all of the terms on the inside of the bracket (feel free to use arrows to help you multiply out the bracket correctly).
It is always easier to work with a positive number of the unknowns; currently we have -3d. Therefore, it is best to move the -3d from the right side of the equation to the left. We are able to do this by adding 3d to both sides of the equation.
The next step is to move the 9 from the left side of the equation to the right. We are able to do this by taking 9 from both sides of the equation.
We want to find the value of d and not 3d. Therefore, we divide both sides of the equation by 3.
This tells us that d is -7.
End Note
Questions like the two examples that we have looked at are perfect examples of where you are given a question that is not in the most straightforward/ typical way. When this is the case, you should think about the formulas/ processes that you would use to find what you are given (for these examples, it was working out the gradient of a line from two points). You then write the formula down and sub in all of the information that you are given, including the unknown. The final step is to solve the equation to find the value of the unknown that you are looking for.
Questions like the two examples that we have looked at are perfect examples of where you are given a question that is not in the most straightforward/ typical way. When this is the case, you should think about the formulas/ processes that you would use to find what you are given (for these examples, it was working out the gradient of a line from two points). You then write the formula down and sub in all of the information that you are given, including the unknown. The final step is to solve the equation to find the value of the unknown that you are looking for.