![]() ![]() ![]() We still don't have an answer for that one. Well, except for that one dream where our hands are giant meatballs. Since the point (3, 14) is indeed on both lines, it's the solution to the system of equations and the answer to all our dreams. Which agrees with the left-hand side of the equation. Is the point (3, 14) on the line y = 3 x + 5? When x = 3 and y = 14, the right-hand side of this equation is Is the point (3, 14) on the line y = 6 x – 4? If it fails either test, we can toss it out with yesterday's garbage. To confirm this, we need to make sure this point satisfies both of the original equations. We think the point (3, 14) is the answer. To find y, we take our value for x, stick it into either equation we like, and solve for y. Until we know y, all we have is half a point, and it's difficult to win an argument with one of those. Since a solution to a system of linear equations is a point, we need to know what y is. Then add 4 to both sides and divide by 3: Start by subtracting 3 x from both sides: So we can substitute (6 x – 4) for y in the second equation: Don't you love it when someone's already come by and done the work for you? Shmoop Algebra: we're a river to our people. The first thing we need to do has already been done: the first equation has been solved for y. Let's do a couple of examples and see what happens. It's exactly the same as when a basketball team makes a substitution, except with less basketball and more math. Even if your textbook didn’t ask you to check your work it’s always a good idea to do this process, it only takes about a minute maybe less and that way you’re going to make sure you get A pluses on your homework and also on your tests.To solve linear systems by substitution, we solve one equation for one variable and then use that information to solve the other equation for the other variable. ![]() That’s how I know I did this problem correctly. My y quantity I hope is equal to 2 times my x quantity take away 3. I also need to check it into the second equation. Good so I’m half way there, I think it's right. SUBSTITUTION EQUATION SYSTEMS PLUSSo here comes my check, first I’m going to check it in the first equation, is it true that my y quantity is equal to 3 times my x quantity plus 1? Let’s see -11 equals -12 plus 1, yap that’s true. In order to check my work, I’m going to go back and plug in -4 for x and -11 to y into both original equations and make sure I get equalities. I’m pretty sure that’s my answer, I’m pretty sure that’s the point where these lines cross even though I didn’t graph them. Y is equal to 3 times my x quantity plus 1, so y is equal to -11 oops, that’s a +1 right there, plus 1 okay. I could also use the second one and I’ll still get the same answer for y. I’m just going to choose to use the first equation. In order to find y, I’m going to take x equals -4, and substitute it into either original equation, that way I’ll get my y value, and I’ll go back and check in a second. I’m going to put a box around this like I would have with my whole answer keeping in mind I still need to find what that y value is. Keep in mind that’s only going to be half of my answer. Now I need to get x all by itself by subtracting 1 from both sides, x is equal to -4. Here we go if I want to find x, I’m going to subtract 2 Xs from both sides, so now I have x plus 1 equals -3. I’m going to find x, and then go back and find my y value. ![]() Now this is a straight forward solving problem. That guy is equal to y and that guy is also equal to y, so I’m just substituting those two equations so they look like one equation with one variable. Since I know y is equal to the expression 3x plus 1, and y is also equal to the expression 2x minus 3, it makes sense mathematically to write 3x plus 1 equals 2x minus 3. This problem using substitution is going to be not too difficult because I have 2 equations that are both already solved for y. ![]()
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