solving systems by graphing goalsgoals study systems of equations study systems of equations solve...
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Solving Systems by Solving Systems by
GraphingGraphing
• GoalsGoals Study systems of equationsStudy systems of equations Solve systems of equations by graphingSolve systems of equations by graphing
• Study checking solutionsStudy checking solutions
Study systems with no solutionStudy systems with no solution Study systems with infinitely many solutionsStudy systems with infinitely many solutions
Systems of EquationsSystems of Equations• Many problems involve more than one Many problems involve more than one
unknown quantity and can be solved using a unknown quantity and can be solved using a system of equations.system of equations.• A A system of linear equationssystem of linear equations consists of two consists of two
linear equations containing two related linear equations containing two related variables.variables.
• A solution to a system of equations is an A solution to a system of equations is an ordered pair (ordered pair (xx, , yy) that satisfies both equations.) that satisfies both equations.
• In this section, systems of linear equations In this section, systems of linear equations will be solved by graphing.will be solved by graphing.
Example 1Example 1
• Find the solution of the following Find the solution of the following system of equations graphically.system of equations graphically.
3 7
3 1
x y
y x
Example 1, cont’dExample 1, cont’d
• Solution: Use a table for each Solution: Use a table for each equation to find at least three points equation to find at least three points for each line.for each line.
• In this example, we could use the In this example, we could use the values values xx = 0, = 0, xx = 1, and = 1, and yy = 0. = 0.
Example 1, cont’dExample 1, cont’d• Solution, cont’d: Find the coordinate Solution, cont’d: Find the coordinate
points on the first line: points on the first line: 3 7x y
Example 1, cont’dExample 1, cont’d• Solution, cont’d: Find the coordinate Solution, cont’d: Find the coordinate
points on the second line: points on the second line: 3 1y x
Example 1, cont’dExample 1, cont’d• Solution, cont’d: Use the points you Solution, cont’d: Use the points you
found to graph the two lines in the found to graph the two lines in the same window.same window.
Example 1, cont’dExample 1, cont’d
• Solution, cont’d: The two lines intersect at Solution, cont’d: The two lines intersect at the point (1, 2), so this is the solution.the point (1, 2), so this is the solution.• Always check the solution in both of the Always check the solution in both of the
original equations.original equations.
Example 2Example 2
• Check that the ordered pair (1, 2) is Check that the ordered pair (1, 2) is the correct solution of the system of the correct solution of the system of equations in Example 1.equations in Example 1.
3 7
3 1
x y
y x
Example 2, cont’dExample 2, cont’d
• Solution: Substitute the point (1, 2) Solution: Substitute the point (1, 2) into the first equation.into the first equation.
Example 2, cont’dExample 2, cont’d• Solution, cont’d: Substitute the point (1, 2) Solution, cont’d: Substitute the point (1, 2)
into the second equation.into the second equation.
• Since the ordered pair checks in both Since the ordered pair checks in both equations, it is the solution.equations, it is the solution.
Systems of Equations, Systems of Equations,
cont’dcont’d• When a system of two linear equations When a system of two linear equations
is graphed, three different outcomes is graphed, three different outcomes are possible.are possible.• The lines may intersect in one point.The lines may intersect in one point.
• The system is called The system is called consistentconsistent..
• The lines may be parallel.The lines may be parallel.• The system is called The system is called inconsistentinconsistent..
• The lines may coincide.The lines may coincide.• The system is called The system is called dependentdependent..
Systems of Equations, Systems of Equations,
cont’dcont’d
• Lines that intersect in one point have different Lines that intersect in one point have different slopes.slopes.
• Parallel or coinciding lines have the same slope.Parallel or coinciding lines have the same slope.
Example 3Example 3
• Solve the system of linear equations Solve the system of linear equations graphically.graphically.
2 4 8
2 2
x y
x y
Example 3, cont’dExample 3, cont’d
• Solution: Put both equations into slope-Solution: Put both equations into slope-intercept form.intercept form.
• The first line has a slope of -½ and a The first line has a slope of -½ and a yy--intercept of 2.intercept of 2.
Example 3, cont’dExample 3, cont’d
• Solution, cont’d:Solution, cont’d:
• The second line has a slope of -½ and The second line has a slope of -½ and a a yy-intercept of -1.-intercept of -1.
Example 3, cont’dExample 3, cont’d
• Solution, cont’d: Since the two lines Solution, cont’d: Since the two lines have the same slope, the direction of have the same slope, the direction of the lines will be the same.the lines will be the same.
• Because the Because the yy-intercepts are different, -intercepts are different, the lines are not identical but must be the lines are not identical but must be parallel.parallel.
• There is no solution.There is no solution.
Example 3, cont’dExample 3, cont’d• Solution, cont’d: Verify this by graphing.Solution, cont’d: Verify this by graphing.
Example 4Example 4
• Solve the system of linear equations Solve the system of linear equations graphically.graphically.
3 2 8
6 4 16
x y
x y
Example 4, cont’dExample 4, cont’d
• Solution: Put both equations into slope-Solution: Put both equations into slope-intercept form.intercept form.
• The first line has a slope of 3/2 and a The first line has a slope of 3/2 and a yy--intercept of -4.intercept of -4.
Example 4, cont’dExample 4, cont’d
• Solution, cont’d:Solution, cont’d:
• The second line also has a slope of 3/2 The second line also has a slope of 3/2 and a and a yy-intercept of -4.-intercept of -4.
Example 4, cont’dExample 4, cont’d
• Solution, cont’d: Since the two lines Solution, cont’d: Since the two lines have the same slope, the direction of have the same slope, the direction of the lines will be the same.the lines will be the same.
• Because the Because the yy-intercepts are also the -intercepts are also the same, the lines are identical.same, the lines are identical.
• There are infinitely many solutions.There are infinitely many solutions.
Example 4, cont’dExample 4, cont’d• Solution, cont’d: Check this by graphing.Solution, cont’d: Check this by graphing.
Example 5Example 5
• Solve the system of linear equations Solve the system of linear equations using a graphing calculator.using a graphing calculator.
4 3 24
2 2
x y
x y
Example 5, cont’dExample 5, cont’d
• Solution: Solve each equation for Solution: Solve each equation for yy..
Example 5, cont’dExample 5, cont’d
• Solution, cont’d: Graph the two Solution, cont’d: Graph the two equations in the same window on the equations in the same window on the calculator.calculator.
Example 5, cont’dExample 5, cont’d
• Solution, cont’d: Use the Intersect to Solution, cont’d: Use the Intersect to calculate the intersection point of (3, 4).calculate the intersection point of (3, 4).