5-1 Graphing Systems of Equations Algebra 1 Glencoe McGraw-Hill Linda Stamper

Notes to copy in my notebook! Two or more linear equations in the same variable form a system of linear equations, or simply a linear system. Here is an example: x + y = 5 Equation 1 2x – 3y = 3 Equation 2 A solution of a linear system in two variables is an ordered pair that makes each equation a true statement. The ordered pair solution is the point of intersection on the graph of the equations. A system of two linear equations can have no solution, one solution, or infinitely many solutions. Notes to copy in my notebook!

When there is one solution it is the point where the graph of each equation intersect. y x Point of intersection (–1,–3) • Intersecting lines – exactly one solution!

If the graphs are parallel, the system of equations has no solution. x Parallel lines will never intersect thus no solution.

Same lines – infinitely many solutions. If the graphs are the same line then there are infinitely many solutions. y x Same lines – infinitely many solutions.

Use the graph to find the solution of the linear system Use the graph to find the solution of the linear system. Then check your solution algebraically using substitution. y   x • The ordered pair (2,–1) makes each equation true, therefore it is the solution of the system of linear equations. Point of intersection (2,–1) (x,y)

Example 1 Is the ordered pair (0,1) a solution of the given system Example 1 Is the ordered pair (0,1) a solution of the given system? Show your work. not a solution  Example 2 Is the ordered pair (-1,2) a solution of the given system? Show your work. solution   If the ordered pair makes each equation true, it is the solution of the system of linear equations.

More notes to copy in my notebook! Solving A Linear System Using Graph–and–Check 1) Write each equation in a form that is easy to graph. 2) Graph both equations in the same coordinate plane. 3) Estimate the coordinate of the point of intersection. 4) Check whether the coordinate gives a solution by substituting it into each equation of the original linear system. More notes to copy in my notebook!

Graph to estimate the solution of the linear system Graph to estimate the solution of the linear system. Then check your solution algebraically using substitution. y –3x + y = 4 –x + 2y = –2 • • • x • slope: 3 y-intercept: 4 • Point of intersection (–2,–2) (x,y)  

Example 3 Graph to estimate the solution of the linear system Example 3 Graph to estimate the solution of the linear system. Then check your solution algebraically using substitution. y • x + y = – 2 2x – 3y = – 9 • • slope: –1 y-intercept: –2 x • • x + y = – 2 (–3) + (1) = – 2 – 2 = – 2 2x – 3y = – 9 2(–3) – 3(1) = – 9 – 6 – 3 = – 9 – 9 = – 9 Point of intersection (–3,1) (x,y)  

Graph to estimate the solution of the linear system Graph to estimate the solution of the linear system. Then check your solution algebraically using substitution. Example 4 x + y = 4 Example 5 x – y = 5 2x + y = 5 2x + 3y = 0 Example 7 Example 6 x – y = –2 x + y = –4

Example 4 Graph to estimate the solution of the linear system Example 4 Graph to estimate the solution of the linear system. Then check your solution algebraically using substitution. y • • x + y = 4 2x + y = 5 • • • slope: –1 y – intercept: 4 x slope: –2 y – intercept: 5 x + y = 4 (1) + (3) = 4 4 = 4 2x + y = 5 2(1) + (3) = 5 2 + 3 = 5 5 = 5   Point of intersection (1,3) (x,y)

Example 5 Graph to estimate the solution of the linear system Example 5 Graph to estimate the solution of the linear system. Then check your solution algebraically using substitution. y x – y = 5 2x + 3y = 0 • x • • slope: 1 y-intercept: – 5 • • Point of intersection (3,–2) (x,y) x – y = 5 (3) – (–2) = 5 5 = 5 2x + 3y = 0 2(3) + 3(–2) = 0 6 – 6 = 0 0 = 0  

Example 6 Graph to estimate the solution of the linear system Example 6 Graph to estimate the solution of the linear system. Then check your solution algebraically using substitution. y x – y = –2 x + y = –4 • • x • • slope: 1 y-intercept: 2 • x – y = – 2 (–3) – (–1) = – 2 – 2 = – 2 x + y = – 4 (–3) + (–1) = – 4 – 4 = – 4 Point of intersection (–3,–1) (x,y)  

Example 7 Graph to estimate the solution of the linear system Example 7 Graph to estimate the solution of the linear system. Then check your solution algebraically using substitution. y • x   Point of intersection (2,–1) (x,y)

Homework 5-1 Page 256-258 # 10-15,17-27odd,28,29,44,45.