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

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

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

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

infinitely many solutions IDENTIFYING THE NUMBER OF SOLUTIONS NUMBER OF SOLUTIONS OF A LINEAR SYSTEM y x y x y x Lines intersect one solution Lines are parallel no solution Lines coincide infinitely many solutions

Solve by Graphing: x + 2y = 7 x + y = 1

Y X (-5,6) Notice the intersection is at a single point.

Name the Solution

Name the Solution

Name the Solution

Intersecting Lines y = x + 5 y = -x -1 2 = -3 + 5 2 = -(-3) – 1 2 = 2  2 = -(-3) – 1 2 = 3 – 1 2 = 2  This screen shows two lines which have exactly one point in common. The common point when substituted into the equation of each line makes that equation true. The common point is (-3,2). Try it and see.

Intersecting Lines y = x + 5 y = -x -1 2 = -3 + 5 2 = -(-3) – 1 2 = 2  2 = -(-3) – 1 2 = 3 – 1 2 = 2  This system of equations is called consistent because it has at least one ordered pair that satisfies both equations. A system of equations that has exactly one solution is called independent.

Coinciding Graphs This graph crosses the y-axis at 2 and has a slope of –2 (down two and right one). Thus the equation of this line is y = -2x + 2. What do you notice about the graph on the right? It appears to be the same as the graph on the left and would also have the equation y = -2x + 2.

Parallel Lines Line 1 crosses the y-axis at 3 and has a slope of 3. Therefore, the equation of line 1 is y = 3x + 3. Line 2 crosses the y-axis at 6 and has a slope of 3. Therefore, the equation of line 2 is y = 3x + 6. Parallel lines have the same slope, in this case 3. However, parallel lines have different y-intercepts. In our example, one y-intercept is at 3 and the other y-intercept is at 6. Parallel lines never intersect. Therefore parallel lines have no points in common and are called inconsistent.

Solve a system of two linear equations in two variables graphically. Key Skills Solve a system of two linear equations in two variables graphically. x y y = 2x - 1 -1 2 3 y = - - 1 2 x + 4 x y 4 2 3

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

Solve a system of two linear equations in two variables graphically. Key Skills Solve a system of two linear equations in two variables graphically. x y y + 2x = 2 2 1 x y y + x = 1 1 1

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