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There are always three ways to solve a system of equationsThere are three ways to solve systems of linear equations: substitution, elimination, and graphing. Let’s review the steps for each method. Substitution
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Graphing
Solving the same system with substitution, then with elimination, then with graphingTake the courseWant to learn more about Algebra 1? I have a step-by-step course for that. :)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. Get access to the complete Algebra 1 courseLearn mathMay 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 5 steps in solving equations by substitution?Steps to Solving by Substitution:. Step One→ Solve one equation for either x or y.. Step Two→ Substitute the expression from step one into the 2nd equation.. Step Three→ Solve the second equation for the given variable.. Step Four→ Plug you solution back into the first equation.. Step Five→ Write your solution as a point.. How do you solve a system of equations by substitution and elimination?To Solve a System of Equations by Elimination. Write both equations in standard form. ... . Make the coefficients of one variable opposites. ... . Add the equations resulting from Step 2 to eliminate one variable.. Solve for the remaining variable.. Substitute the solution from Step 4 into one of the original equations.. How do you solve two equations using substitution?To solve systems using substitution, follow this procedure:. Select one equation and solve it for one of its variables.. In the other equation, substitute for the variable just solved.. Solve the new equation.. Substitute the value found into any equation involving both variables and solve for the other variable.. What is an example of a substitution equation?If you substitute the values x = −3 and y = 2 into the first equation, you get a false statement: 2(2) = −3 + 9. To solve this system, try rewriting the first equation as x = 2y − 8. Then substitute 2y − 8 in for x in the second equation, and solve for y. The correct answer is x = −2, y = 3.
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