Determine the system of equations illustrated in each graph. W...

Question

Determine the system of equations illustrated in each graph. Write equations in standard form.

Graph showing two intersecting lines with points labeled (5/2, 1/2) and (2, 3) on the coordinate plane.

Accepted Answer

Hey, everyone in this video, we're gonna be working through the following problem. So we're asked in standard form to write the system of equations associated with the graph show the graph shown has two lines. OK that intersect at one point and we're giving four points. So we're given two points on each line. The first line has the points 0 10 and negative 40. The second line has the points negative 80 and zero, negative three. We're given four answer choices A through D, each of them just containing a different combination of equations for these two lines. Let's go back up to our graph where we can see those points. We're gonna start with the first equation. We're gonna call it Y one and that's gonna be the equation with the points zero, comma 10 and negative four comma zero. Now, let's recall, we can write the equation of a line as Y is equal to M X plus B. And in this case, we're gonna use a subscript one. So we have Y one is equal to M one, X one plus B one and the points were given zero comma 10 and negative four comma zero. And you could also use the point slope form of a line as well. Um To, to do this problem, it would give you the exact same result and it's about the same amount of work. So whichever one you're more comfortable with. Now, if we take a look at this equation or sorry, not the equation, the line, the graph and the points we're given, we can see that the Y intercept, OK is equal to when we have our equation and Y is equal to M X plus B, we're called that B represents A Y intercept. So what that means is that B one must be equal to. OK? And you can see this if you forget that B one is A Y intercept, you can substitute in that point as well. OK? If you substitute X equals zero, you get Y one is equal to B one and the Y value when X is equal to zero is 10. So you can substitute that point in as well. If you forget that our equation is going to be Y one is equal to M one, X one. What's that? And we're gonna use our other point now to find what M should be. So we're gonna substitute negative four comma zero into our equation. You get that zero is equal to M multiplied by negative four. What? OK? We can move the 10 to the left hand side by subtracting. And then we divide by negative four to isolate for M, we forget that M is equal to negative 10, divided by negative four. The new writer and denominator are both divisible by two. So we can simplify and write this as five halves, right? So we have our M one value. So there should be a subscript one there. So we have our M one value. So we can write that Y is equal to five halves X bluster. Now, we're asked to write this in standard form and we're called that standard form of a line. We wanna have the Y and X terms on one side and the constant term on the other. So what we're gonna do is we're gonna start by multiplying the entire equation by two to get rid of this fraction because there are no fractions in our answer choices. If we do that, we have two, Y is equal to five X plus one. And we're gonna move the five X to the left hand side by subtracting it in order to get this in standard form. And we get that two Y minus five X is equal to 20. So that's the first equation we have in our system. If we take a look at our answer choices, OK, we can see that we can already eliminate some of them. OK? So for option A, the first equation is correct for option B and C it's not OK. There is a negative two Y instead of positive two, Y on the left hand side. And so those equations are incorrect. So we're down to option A or D and now we're gonna flip over to that second line and figure out the equation. So flipping over to the second line, we have the points negative eight comma zero and zero, comma negative three. And we're gonna do the exact same thing. So we have Y two is equal to M two, X two plus B two. And we have the points negative eight comma zero and zero, comma negative three. So just like the first one, if we have a Y intercept given when X is equal to zero, which is negative three. And so our B value in this case, B two is going to be equal to negative three. Now we're gonna substitute the other point which is negative eight comma zero. You get that Y which is zero is equal to M two, multiplied by X which is negative eight minus three. We can move the three to the left hand side by adding it and then divide by negative eight to isolate for M two. We get that M two is equal to negative three divided by eight. And so we can write our equation as Y is equal to negative 38, X minus three. Once again, we want this in standard form. So we're gonna have to multiply and rearrange, we're gonna multiply both sides of the equation by eight, we get that eight Y is equal to negative three X minus 24. And moving the three X to the other side, we have that eight Y plus three X is equal to negative 24. And that is the second equation. So if we go up to our answer choices, we had narrowed it down to A and D, we found that the second equation should be three X plus eight Y is equal to negative 24. And so we have option A. Thanks everyone for watching. I hope this video helped see you in the next one.

Determine the system of equations illustrated in each graph. Write equations in standard form.

Key Concepts

Finding the Equation of a Line from Intercepts

Converting to Standard Form of a Linear Equation

Interpreting Graphs to Identify Systems of Equations

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