Video: US-SAT05S4-Q11-179125620415

Which of the following is the equation of a line that when graphed in the π‘₯𝑦-plane has a slope of 3 and contains the point (1, 5)? [A] 3π‘₯ βˆ’ 𝑦 = βˆ’2 [B] 3π‘₯ + 𝑦 = βˆ’2 [C] 3π‘₯ βˆ’ 𝑦 = 8 [D] 3π‘₯ + 𝑦 = 8

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Video Transcript

Which of the following is the equation of a line that when graphed in the π‘₯𝑦-plane has a slope of three and contains the point one, five? Is it A) three π‘₯ minus 𝑦 equals negative two, B) three π‘₯ plus 𝑦 equals negative two, C) three π‘₯ minus 𝑦 equals eight, or D) three π‘₯ plus 𝑦 equals eight?

Let’s firstly consider the general equation of a straight line in slope intercept form. This is written 𝑦 equals π‘šπ‘₯ plus 𝑏. The value of π‘š is the slope or gradient of the line, and the value of 𝑏 is the 𝑦-intercept. In this question, we’re looking for an equation which has a slope or gradient of three. We can work out which of the four options have a slope of three by rearranging them in the form 𝑦 equals π‘šπ‘₯ plus 𝑏.

Let’s firstly consider option A, three π‘₯ minus 𝑦 is equal to negative two. Our first step is to subtract three π‘₯ from both sides of the equation. This gives us negative 𝑦 is equal to negative three π‘₯ minus two. As we want 𝑦 on its own on the left-hand side, we need to divide by negative one. Dividing the left-hand side by negative one means we have to divide the right-hand side by negative one. This means that the equation simplifies to 𝑦 equals three π‘₯ plus two. This equation does have a slope of three. So it could be the right answer.

We now need to repeat this process for option B, three π‘₯ plus 𝑦 equals negative two. Once again, we need to subtract three π‘₯ from both sides of this equation. This gives us 𝑦 is equal to negative three π‘₯ minus two. This time, the slope or gradient is equal to negative three. So this cannot be the correct answer. We’ve therefore eliminated option B.

Option C, three π‘₯ minus 𝑦 is equal to eight, can also be rearranged in the form 𝑦 equals π‘šπ‘₯ plus 𝑏. Subtracting three π‘₯ from both sides and then dividing by negative one gives us 𝑦 is equal to three π‘₯ minus eight. This equation does have a slope of three. So once again, could be the correct answer.

Our final equation was three π‘₯ plus 𝑦 is equal to eight. Subtracting three π‘₯ from both sides of this equation gives us 𝑦 is equal to negative three π‘₯ plus eight. As was the case with option B, this has a slope of negative three. So once again, cannot be the correct answer.

We know that either option A or option C is the correct answer and must now use the second piece of information. We’re told that the line passes through the point one, five. This means that when we substitute in these coordinates, the left-hand side will be equal to the right-hand side. Substituting in the coordinate or point one, five gives us five is equal to three multiplied by one plus two for option A.

For option C, we get five is equal to three multiplied by one minus eight. Three multiplied by one is equal to three. So five is equal to three plus two. As the left-hand side and right-hand side are both equal to five, this equation does pass through the point one, five. With option C, we have three minus eight on the right-hand side. This is equal to negative five, which is not equal to five. Option C, three π‘₯ minus 𝑦 equals eight, does have a slope of three. But it does not pass through the point one, five.

We can therefore conclude that the correct equation is option A, three π‘₯ minus 𝑦 equals negative two. This has a slope of three and contains the point one, five. We could sketch this on the π‘₯𝑦-plane. The line with a slope of three that passes through the point one, five intercepts the 𝑦-axis at two. This confirms that our rearranged equation in the form 𝑦 equals π‘šπ‘₯ plus 𝑏 is correct.

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