Solving Systems of Linear Equations

Wait a minute! What’s a system of linear equations? –A system is a set of linear sentences which together describe a single situation. How do I know if I have a solution? –A solution to a system is a pair of numbers that satisfies all equations within the solution.

Examples of Linear Systems The system at the upper left shows an example of two linear equations in standard form (Ax+By=C). The system at the lower left shows an example of two linear equations in slope-intercept form (y=mx+b).

Non-examples of Linear Systems The situation in the upper left is a system of equations, but it is not a system of linear equations because the top equation is a quadratic. The situation at the lower left is not a system because it is not a set of linear sentences. It is only one linear sentence.

Which of the following is a system of linear equations? *Note: The equations are brought in as objects from Equation Editor. Therefore, press the button in the blue space below the equations.

Be Careful… This is a system of equations, but both equations are not linear. Linear equations always have x raised to the first power. Review Examples of Linear Systems

Great Job! Now that you can identify a system of linear equations, it is time to learn how to solve them!

Solving Systems of Linear Equations Solving by Graphing Solving by Substitution Solving by Elimination

Steps to Solve by Graphing 1.Graph both equations in the system. 2.Find the ordered pair at the point of intersection. 3.This ordered pair is the solution to the system! (It contains the x and y values that will make both equations true!) Always check your solution in the equations.

Important Note: It is VERY important to graph your lines accurately! Use graph paper and a straight edge! Let’s look at an example!

Solve First, graph y=2x+1. Remember, plot the y- intercept first (1) and then use the slope (2) to find another point on the line.

Solve Second, graph y=-3x+6 on the same set of axes.

The solution is the point of intersection! The lines intersect at (1,3). This means that when x=1, then y=3 in BOTH equations in the system. To be sure, our next step is to check the solution in the equations.

Does (1,3) work in both equations? Let’s check!

(1,3) is the solution to Since (1,3) is the point of intersection and it worked in both equations of the system, this is the solution! Test your understanding.

Solve The first step is to graph both of these equations.

Solve Choose the graph below that has both equations graphed correctly. (Click in the yellow area below the graph!)

Now check your solution! The intersection point is (-1,1). Does this work in both equations? The solution does not work in either equation! This means the graph is not correct!

Now check your solution! The intersection point is (-2,-2). Does this work in both equations? The solution does not work in the second equation! This means the graph is not correct!

Now check your solution! The intersection point is (-3,-3). Does this work in both equations? Great job! (-3,-3) works in both equations so it is the solution!

Solving by Substitution Look at the system We can say that

Solving by Graphing You have done a great job solving systems of equations by graphing. There are two other techniques to solving systems of linear equations. –Solving by Substitution –Solving by Elimination Return to the Home Menu to learn more!

By the Transitive Property… The Transitive Property states that if a=b and b=c, then a=c. From our system, we know that 5x-25=y and y=-8x+27. Therefore, 5x-25=-8x+27! –This is an equation with one variable. We can solve for x!

Solve the Equation

X=4 is part of the solution! When x=4, both equations will result in the same y value. This is the other coordinate in our solution! Substitute x=4 into one of the equations. (It does not matter which equation you use; both will give the same result for y.)

Let’s find the y value! Regardless of which equation you use, when x=4 then y=-5. Therefore, the solution to this system is (4,-5).

Rules for Solving by Substitution 1.Solve both equations for y. 2.Set the equations equal to each other. 3.Solve for x. 4.Substitute the x value into one of the equations to solve for y. 5.Once you have x and y, write your solution as an ordered pair.

Try this example. Using substitution, solve Set the equations equal to each other and solve for x. When you do this, what result do you get? X=3X=5X=12

If x=3, solve for y. Does the solution (3,-3.5) check in both equations?

Check the solution (3,-3.5) (3,-3.5) does not work in the second equation. Therefore, this is not the solution.

If x=5, solve for y. Does the solution (5,-2.5) check in both equations?

Check the solution (5,-2.5) (5,-2.5) does not work in the second equation. Therefore, this is not the solution.

If x=12, solve for y. Does the solution (12,1) check in both equations?

Check the solution (12,1) Great job! Since (12,1) works in both equations, it is the solution to this system!

Solving by Substitution You have done a great job solving systems of equations by substitution. There are two other techniques to solving systems of linear equations. –Solving by Graphing –Solving by Elimination Return to the Home Menu to learn more!

Solving by Elimination We know that if a=b and c=d, then a+c=b+d. –Apply this to equations to solve by elimination. The goal is to add the equations of a system to get a variable to cancel out.

Solve First, add this equations together. Since there is –y in the first equation and +y in the second equation the y variable will cancel out! Now that we know what x is, we can solve for y!

If x=135, solve for y! You can use either equation to solve for y. Our solution is (135,15). To verify that this is the solution, we can check the coordinates in both equations.

Check the solution (135,15) Since (135,15) works in both equations, it is the solution to this linear system! Now try an elimination problem on your own…

Solve When you add the equations in the system together, which variable will you end up solving for? xy

Check again… When you add the equations, 2x+-2x will cancel out, leaving you y to solve for.

Good job! Now solve for y! What is the result? Y=2/3Y=1Y=2

If y=2/3, solve for x. Does the solution check in both equations?

Check the solution This ordered pair does not work in the second equation. Therefore, it is not the solution to the system.

If y=1, solve for x. Does the solution (-3,1) check in both equations?

Check the solution (-3,1) This ordered pair does not work in the second equation. Therefore, it is not the solution to the system.

If y=2, solve for x. Does the solution (-7,2) check in both equations?

Check the solution (-7,2) Good job! This ordered pair works in both equations. Therefore, (-7,2) is the solution to the system!

Solving by Elimination You have done a great job solving systems of equations by elimination. There are two other techniques to solving systems of linear equations. –Solving by Graphing –Solving by Substitution Return to the Home Menu to learn more! If you have already learned all three methods, click on the blue arrow for more information!

Solving Systems of Equations You have now learned the basic principles to solving systems of linear equations using the three methods: graphing, substitution, and elimination. For more practice on solving systems of linear equations (or to look at more advanced examples) click here.here Back to the beginning… End