Solving Systems by Graphing GoalsGoals Study systems of equations Study systems of equations Solve systems of equations by graphing Solve systems of equations by graphing Study checking solutionsStudy checking solutions Study systems with no solution Study systems with no solution Study systems with infinitely many solutions Study systems with infinitely many solutions

Systems of Equations Many problems involve more than one unknown quantity and can be solved using a system of equations.Many problems involve more than one unknown quantity and can be solved using a system of equations. A system of linear equations consists of two linear equations containing two related variables.A system of linear equations consists of two linear equations containing two related variables. A solution to a system of equations is an ordered pair (x, y) that satisfies both equations.A solution to a system of equations is an ordered pair (x, y) that satisfies both equations. In this section, systems of linear equations will be solved by graphing.In this section, systems of linear equations will be solved by graphing.

Example 1 Find the solution of the following system of equations graphically.Find the solution of the following system of equations graphically.

Example 1, cont’d Solution: Use a table for each equation to find at least three points for each line.Solution: Use a table for each equation to find at least three points for each line. In this example, we could use the values x = 0, x = 1, and y = 0.In this example, we could use the values x = 0, x = 1, and y = 0.

Example 1, cont’d Solution, cont’d: Find the coordinate points on the first line:Solution, cont’d: Find the coordinate points on the first line:

Example 1, cont’d Solution, cont’d: Find the coordinate points on the second line:Solution, cont’d: Find the coordinate points on the second line:

Example 1, cont’d Solution, cont’d: Use the points you found to graph the two lines in the same window.Solution, cont’d: Use the points you found to graph the two lines in the same window.

Example 1, cont’d Solution, cont’d: The two lines intersect at the point (1, 2), so this is the solution.Solution, cont’d: The two lines intersect at the point (1, 2), so this is the solution. Always check the solution in both of the original equations.Always check the solution in both of the original equations.

Example 2 Check that the ordered pair (1, 2) is the correct solution of the system of equations in Example 1.Check that the ordered pair (1, 2) is the correct solution of the system of equations in Example 1.

Example 2, cont’d Solution: Substitute the point (1, 2) into the first equation.Solution: Substitute the point (1, 2) into the first equation.

Example 2, cont’d Solution, cont’d: Substitute the point (1, 2) into the second equation.Solution, cont’d: Substitute the point (1, 2) into the second equation. Since the ordered pair checks in both equations, it is the solution.Since the ordered pair checks in both equations, it is the solution.

Systems of Equations, cont’d When a system of two linear equations is graphed, three different outcomes are possible.When a system of two linear equations is graphed, three different outcomes are possible. The lines may intersect in one point.The lines may intersect in one point. The system is called consistent.The system is called consistent. The lines may be parallel.The lines may be parallel. The system is called inconsistent.The system is called inconsistent. The lines may coincide.The lines may coincide. The system is called dependent.The system is called dependent.

Systems of Equations, cont’d Lines that intersect in one point have different slopes.Lines that intersect in one point have different slopes. Parallel or coinciding lines have the same slope.Parallel or coinciding lines have the same slope.

Example 3 Solve the system of linear equations graphically.Solve the system of linear equations graphically.

Example 3, cont’d Solution: Put both equations into slope- intercept form.Solution: Put both equations into slope- intercept form. The first line has a slope of -½ and a y- intercept of 2.The first line has a slope of -½ and a y- intercept of 2.

Example 3, cont’d Solution, cont’d:Solution, cont’d: The second line has a slope of -½ and a y-intercept of -1.The second line has a slope of -½ and a y-intercept of -1.

Example 3, cont’d Solution, cont’d: Since the two lines have the same slope, the direction of the lines will be the same.Solution, cont’d: Since the two lines have the same slope, the direction of the lines will be the same. Because the y-intercepts are different, the lines are not identical but must be parallel.Because the y-intercepts are different, the lines are not identical but must be parallel. There is no solution.There is no solution.

Example 3, cont’d Solution, cont’d: Verify this by graphing.Solution, cont’d: Verify this by graphing.

Example 4 Solve the system of linear equations graphically.Solve the system of linear equations graphically.

Example 4, cont’d Solution: Put both equations into slope- intercept form.Solution: Put both equations into slope- intercept form. The first line has a slope of 3/2 and a y- intercept of -4.The first line has a slope of 3/2 and a y- intercept of -4.

Example 4, cont’d Solution, cont’d:Solution, cont’d: The second line also has a slope of 3/2 and a y-intercept of -4.The second line also has a slope of 3/2 and a y-intercept of -4.

Example 4, cont’d Solution, cont’d: Since the two lines have the same slope, the direction of the lines will be the same.Solution, cont’d: Since the two lines have the same slope, the direction of the lines will be the same. Because the y-intercepts are also the same, the lines are identical.Because the y-intercepts are also the same, the lines are identical. There are infinitely many solutions.There are infinitely many solutions.

Example 4, cont’d Solution, cont’d: Check this by graphing.Solution, cont’d: Check this by graphing.

Example 5 Solve the system of linear equations using a graphing calculator.Solve the system of linear equations using a graphing calculator.

Example 5, cont’d Solution: Solve each equation for y.Solution: Solve each equation for y.

Example 5, cont’d Solution, cont’d: Graph the two equations in the same window on the calculator.Solution, cont’d: Graph the two equations in the same window on the calculator.

Example 5, cont’d Solution, cont’d: Use the Intersect to calculate the intersection point of (3, 4).Solution, cont’d: Use the Intersect to calculate the intersection point of (3, 4).