Solving Linear Systems by Graphing

Warm-Up Exercises ANSWER y x =+ 3 Evaluate for and 1. 5x5x+2y2y 2x = 4.4. y = – Find the slope-intercept form of the equation x3x+4y4y = – ANSWER 18xC = Express the cost C of x ball game tickets at a price of $18 per ticket. 3.

 System of two linear equations: in two variables x and y consists of two equations. The coefficients of the terms in the equations can be any real numbers.  Solution: of a system of two linear equations in two variables is an ordered pair (x,y) that satisfies both equations. When you graph the system, the solution is represented by the point of intersection of the two lines.

Example 1 Solve a System by Graphing Solve the system by graphing. Then check your solution algebraically. 33x3x–y = 8x+2y2y = Equation 1 Equation 2 SOLUTION Graph both equations, as shown. From the graph, you can see the lines appear to intersect at ().). 2, 32, 3

Example 1 Solve a System by Graphing You can check the solution by substituting 2 for x and 3 for y into the original equations. Equation 1Equation 2 33x3x–y = 8x+ 2y2y = () 233–3 = ? 28 + = ? () 32 36–3 = ? 28 + = ? 6 33 = 88 = ANSWER ().). 2, 3 The solution of the system is

Checkpoint Solve the system by graphing. Then check your solution. 1. 3y–x = + 9 y 2x = + ANSWER ()2, 5 – Solve a System by Graphing

Checkpoint 2. x3y3y = – 1 xy = + 1 –– ANSWER ()1, 0 Solve a System by Graphing Solve the system by graphing. Then check your solution.

Checkpoint 3. = 2x2x3y3y = – 6 ANSWER ()6, 2 Solve a System by Graphing Solve the system by graphing. Then check your solution. 2+4y4yx –

Systems with Many or No Solutions Example 2 Tell how many solutions the linear system has. a. 1 = y – 2x2x+ = 2y2y – 4x4x2 – b. + = 2y2yx1 + = 2y2yx4 a. SOLUTION Because the graph of each equation is the same, each point on the line is a solution. So, the system has infinitely many solutions.

Example 2 b.Because the graphs of the equations are two parallel lines, the two lines have no point of intersection. So, the system has no solution. Systems with Many or No Solutions

Write and Use a Linear System Example 3 Vacation You are planning a 7 -day trip to California. You estimate that it will cost $300 per day in San Diego and $400 per day in Anaheim. Your total budget for the trip is $2400. How many days should you spend in each city? SOLUTION You can use a verbal model to write a system of linear equations. VERBAL MODEL Total vacation time Days in San Diego Days in Anaheim =+ Daily cost in San Diego = + Daily cost in Anaheim Total Budget Days in Anaheim Days in San Diego +

Write and Use a Linear System Example 3 LABELS Days in San Diego x = (days) Days in Anaheim y = (days) Total vacation time 7 = (dollars per day) Daily cost in San Diego 300 = (dollars per day) Daily cost in Anaheim 400 = (days) Total budget 2400 = (dollars) ALGEBRAIC MODEL Equation 1 (total vacation time) 7 = x + y Equation 2 (total budget) 2400 = 300x + 400y

Write and Use a Linear System Example 3 Graph both equations only in the first quadrant because the only values that make sense in this situation are positive values of x and y. The lines appear to intersect at. () 4, 34, 3

Write and Use a Linear System Example 3 CHECK Substitute 4 for x and 3 for y in the original equations. Equation = 300x + 400y = () () 3 Equation 1 = x + y7 = ANSWER The solution is. You should plan to spend 4 days in San Diego and 3 days in Anaheim. () 4, 3

Tell how many solutions the linear system has. Checkpoint ANSWER 0 Write and Use Linear Systems 5. 5 = 4y4y – x + = 4y4y – x5 – 4. + = 3y3y2x2x1 + = 6y6y4x4x3 ANSWER infinitely many solutions ANSWER = 5y5y – x + = 5y5yx 5

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