For these lines to intersect, 2 must equal 8, which is ridiculous. What does this incorrect equation tell us? There's no solution to the system. We know 2 doesn't equal 8, or else the Raptors got royally ripped off by the official scorers at last night's game. When we combine the x terms, we're left with the statement 2 = 8. Even if "Trouble" is your middle name, you're not going to like what comes next. To solve, first we simplify to find x + 2 – x = 8.
We substitute for y in the equation 2 y – x = 8 to get: The first equation is already solved for y, which makes our lives better. How's that for a wide range of options? Just somewhere between "none" and "infinity," that's all. So far, each of the systems we've solved using substitution has had exactly one answer, but a system of equations could have no solutions or infinitely many solutions. How about that it actually worked! There may be a place for fractions in the universe after all. Let's see if 2 y – 3 x really does equal 8 for these bizarro values of x and y. When and, the left-hand side of the equation isĭo these values work in the equation 2 y – 3 x = 8? Do these values work in the equation x + 4 y = 7? Check that the answer works in both original equations. Find the value of the variable we solved for in step 1.Īnother fraction. Where are you from, fraction? Oh, really? Well did you.okay, we can't do this. Let's try to overlook our dislike of fractions, though, and make the most of a bad situation. However, it's the best we can do in this instance. Simplify a bit more to get 14 y = 29, and divide by 14 to track down y: Simplify that thing to find 2 y – 21 + 12 y = 8.
After substituting, solve the other equation. Performing substitution gives us 2 y – 3(7 – 4 y) = 8.ģ. In the other equation, perform substitution to get rid of the variable we solved for in step 1. The first equation has x all by itself (with a coefficient of 1), so it's easiest to solve that equation for x. Check that the answer works in both original equations.Find the value of the variable we solved for in step 1.After substituting, solve the other equation.In the other equation, perform substitution to get rid of the variable we solved for in step 1.It's great that you wanted to build a fort out of the couch cushions, but people have to live here. Once we're done, we should also tidy up the living room. Let's tidy things up a bit and figure out the general steps we need to take for this sort of problem.
We've now used substitution to successfully find the point of intersection for two lines that intersect exactly once. 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. 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.
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