Systems of Equations Solving by Graphing

Systems of Equations One way to solve equations that involve two different variables is by graphing the lines of both equations on a coordinate plane. If the two lines cross the solution for both variables is the coordinate of the point where they intersect.

y = 2x + 0 & y = -1x + 3 Slope = 2/1 y-intercept= 0 Up 2 and right 1 y-intercept= +3 Slope = 1/-1 Up 1 and left 1 The solution is the point they cross at (1,2) (1,2)

y = x - 3 & y = -3x + 1 Slope = 1/1 y-intercept= -3 y-intercept= +1 Slope = 3/-1 The solution is the point they cross at (1,- 2)

The solution is the point they cross at (1,2) y =-2x + 4 & y = 2x + 0 Slope = 2/-1 y-intercept= 4 y-intercept= 0 Slope = 2/1

Graph y = x -3 y = x + 2 Solution= none

N UMBER OF S OLUTIONS OF A L INEAR S YSTEM I DENTIFYING T HE N UMBER OF S OLUTIONS y x y x Lines intersect one solution Lines are parallel no solution y x Lines coincide infinitely many solutions

Change to y = mx + b

y = -1/2x + 7/2 y = -x + 1 (-5,6) 7/2 = 3 1/2 Notice the intersection is at a single point.

Name the Solution

x – 2 = y Step 2:2:Use the slope and y-intercept of each line to plot two points for each line on the same graph. x y Place a point at –2 on the y-axis. Since the slope is 1, move up 1 and right 1 and place another point.

x – 2 = y Step 2:Use the slope and y-intercept of each line to plot two points for each line on the same graph. x y Place a point at 3 on the y- axis for the second line. The second line has a slope of negative 2/3. From the y-intercept, move down two and right 3 and place another point.

x y Step 3:3: Draw in each line on the graph. Step 4: Determine the point of intersection. The point of intersection of the two lines is the point (3,1). This system of equations has one solution, the point (3,1).

Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.

x y The two equations in slope- intercept form are: Plot points for each line. Draw in the lines. These two equations represent the same line. Therefore, this system of equations has infinitely many solutions.

The two equations in slope- intercept form are: x y Plot points for each line. Draw in the lines. This system of equations represents two parallel lines. This system of equations has no solution because these two lines have no points in common.

x y The two equations in slope- intercept form are: Plot points for each line. Draw in the lines. This system of equations represents two intersecting lines. The solution to this system of equations is a single point (3,0).

Key Skills Solve a system of two linear equations in two variables graphically. y = 2x  1 y =  1 2 x –246–4 x y 6 –2 –4 –6 solution: (2, 3)

Key Skills Solve a system of two linear equations in two variables graphically –246–4 x y 6 –2 –4 –6 solution:≈ (1, 0) y + 2x = 2 y + x = 1

Key Skills Solve a system of two linear equations in two variables graphically –246–4 x y 6 –2 –4 –6 No solution, why? y = 2x + 2 y = 2x + 4 Because the 2 lines have the same slope.

Key Skills Solve a system of two linear equations in two variables graphically. y = 3x + 2 y =  1 3 x –246–4 x y 6 –2 –4 –6 solution:≈ (-3, -1) TRY THIS

Key Skills Solve a system of two linear equations in two variables graphically –246–4 x y 6 –2 –4 –6 solution:≈ (-1.5, -3) 2x + 3y = -12 4x – 4y = 4 TRY THIS

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Systems of Equations Solving by Graphing