There are three ways to solve systems of linear equations: substitution, elimination, and graphing. Let’s review the steps for each method. Show
Substitution
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Elimination
Graphing
Solving the same system with substitution, then with elimination, then with graphing
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Determining which method is best for solving the system: substitution, elimination, or graphingNow let’s look at a few examples in which we need to decide which of these three methods to use. Example Which method would you use to solve the following problem? Explain why you picked the method that you did. ???x=y+2??? ???3y-2x=15??? The easiest way to solve this system would be to use substitution since ???x??? is already isolated in the first equation. Whenever one equation is already solved for a variable, substitution will be the quickest and easiest method. Even though you’re not asked to solve, these are the steps to solve the system: Substitute ???y+2??? for ???x??? in the second equation. ???3y-2(y+2)=15??? Distribute the ???-2??? and then combine like terms. ???3y-2y-4=15??? ???y-4=15??? Add ???4??? to both sides. ???y-4+4=15+4??? ???y=19??? Plug ???19??? for ???y??? into the first equation. ???x=y+2??? ???x=19+2??? ???x=21??? The unique solution is ???(21,19)???. There are three ways to solve systems of linear equations: substitution, elimination, and graphing. How to solve a system using the elimination methodExample To solve the system by elimination, what would be a useful first step? ???x+3y=12??? ???2x-y=5??? When we use elimination to solve a system, it means that we’re going to get rid of (eliminate) one of the variables. So we need to be able to add the equations, or subtract one from the other, and in doing so cancel either the ???x???-terms or the ???y???-terms. Any of the following options would be a useful first step: Multiply the first equation by ???-2??? or ???2???. This would give us ???2x??? or ???-2x??? in both equations, which will cause the ???x???-terms to cancel when we add or subtract. Multiply the second equation by ???3??? or ???-3???. This would give us ???3y??? or ???-3y??? in both equations, which will cause the ???y???-terms to cancel when we add or subtract. Divide the second equation by ???2???. This would give us ???x??? or ???-x??? in both equations, which will cause the ???x???-terms to cancel when we add or subtract. Divide the first equation by ???3???. This would give us ???y??? or ???-y??? in both equations, which will cause the ???y???-terms to cancel when we add or subtract. Let’s re-do the last example, but instead of the elimination method, use a graph to find the solution. Solving the system by graphing both equations and finding the intersection pointsExample Graph both equations to find the solution to the system. ???x+3y=12??? ???2x-y=5??? In order to graph these equations, let’s put both of them into slope-intercept form. We get ???x+3y=12??? ???3y=-x+12??? ???y=-\frac13x+4??? and ???2x-y=5??? ???-y=-2x+5??? ???y=2x-5??? The line ???y=-(1/3)x+4??? intersects the ???y???-axis at ???4???, and then has a slope of ???-1/3???, so its graph is The line ???y=2x-5??? intersects the ???y???-axis at ???-5???, and then has a slope of ???2???, so if you add its graph to the graph of ???y=-(1/3)x+4???, you get Looking at the intersection point, it appears as though the solution is approximately ???(3.75,2.75)???. In actuality, the solution is ???(27/7,19/7)\approx(3.86,2.71)???, so our visual estimate of ???(3.75,2.75)??? wasn’t that far off.
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Learn mathKrista KingMay 4, 2019math, learn online, online course, online math, algebra, algebra 1, algebra i, algebra 2, algebra ii, solving systems, solving linear systems, systems of equations, systems of linear equations, substitution, solving with substitution, elimination, solving with elimination, graphing, solving by graphing, solving systems with substitution, solving systems with elimination, solving systems by graphing, substitution method, elimination method What are the 3 types of system of equations?The types of systems of linear equations are as follows:. Dependent: The system has infinitely many solutions. The graphs of the equations represent the same lines. ... . Independent: The system has exactly one solution. The graphs of the equations intersect at a single point. ... . Inconsistent: The system has no solution.. What are the 3 possible cases for solutions to a system of linear equations?In fact there are only three possible cases:. No solution.. One solution.. Infinitely many solutions.. What are the solutions to the system of equations?The solution set to a system of equations will be the coordinates of the ordered pair(s) that satisfy all equations in the system. In other words, those values of x and y will make the equations true. Accordingly, when a system of equations is graphed, the solution will be all points of intersection of the graphs.
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