 # Evaluate each expression for x = 1 and y = –3. 1. x – 4y 2. –2x + y Write each expression in slope-intercept form. 3. y – x = 1 4. 2x + 3y = 6 5. 0 = 5y.

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Evaluate each expression for x = 1 and y = –3. 1. x – 4y 2. –2x + y Write each expression in slope-intercept form. 3. y – x = 1 4. 2x + 3y = 6 5. 0 = 5y + 5x 13 –5 y = x + 1 y = x + 2 y = –x Warm Up

Lesson 6.1 Solving Systems of Equations by Graphing

Objective 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. Also covered: 6.0 California Standards

What is a system of equations? A system of equations is when you have two or more equations using the same variables. The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair (x, y). When graphing, you will encounter three possibilities.

1) Intersecting Lines The point where the lines intersect is your solution. The solution of this graph is (1, 2) (1,2)

What is the solution of the system graphed below? 1. (2, -2) 2. (-2, 2) 3. No solution 4. Infinitely many solutions

2) Parallel Lines These lines never intersect! Since the lines never cross, there is NO SOLUTION! Parallel lines have the same slope with different y-intercepts.

3) Coinciding Lines These lines are the same! Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! Coinciding lines have the same slope and y-intercepts.

What is the solution of this system? y=3x – 8 y = 3x -8 1. (3, 1) 2. (4, 4) 3. No solution 4. Infinitely many solutions

Problem-Solving Application Sally and Jenni are reading the same book. Wren is on page 14 and reads 2 pages every night. Jenni is on page 6 and reads 3 pages every night. After how many nights will they have read the same number of pages? How many pages will that be?

 (8, 30) Nights

Lets try one: Find the solution to the following system: y = -2x+4 y = x -2  Step 1:Graph both equations m=-2 b=4 m=1 b=-2  Step 2: Find Where the lines intersect? The point is (2, 0)

Check your answer! To check your answer, substitute the point back into both equations. Y=-2x+4 2(2) + 4 = 0 y = x-2 (2) – 2 = 0 Nice job…let’s try another!

You try: Find the solution to the following system  Graph both equations  Where do they intersect? y = 2x – 1 y = –x + 5 The point is (2, 3)

Try again: Find the solution to the following system  Graph both equations  Where do they intersect? no solution

Challenge: Find the solution to the following system: y = 2x – 3 -2x + y = 1  Graph both equations.  Put both equations in slope-intercept y=mx+b y = 2x – 3 y = 2x + 1 Graph using slope and y-intercept

Graph the equations. y = 2x – 3 m = 2 and b = -3 y = 2x + 1 m = 2 and b = 1 Where do the lines intersect? No solution! Notice that the slopes are the same with different y-intercepts. If you recognize this early, you don’t have to graph them!

Solving a system of equations by graphing. Let's summarize! There are 3 steps to solving a system using a graph. Step 1: Graph both equations. Step 2: Do the graphs intersect? Step 3: Check your solution. Graph using slope and y – intercept or x- and y-intercepts. Be sure to use a ruler and graph paper! This is the solution! LABEL the solution! Substitute the x and y values into both equations to verify the point is a solution to both equations.

Tell whether the ordered pair is a solution of the given system. Identifying Systems of Solutions (–2, 2); –x + y = 2 x + 3y = 4 Substitute –2 for x and 2 for y. x + 3y = 4 –2 + 6 4 4 4 –2 + (3)2 –x + y = 2 4 2 2 –(–2) + 2  The ordered pair (–2, 2) makes one equation true, but not the other. (–2, 2) is not a solution of the system.

Tell whether the ordered pair is a solution of the given system. (1, 3); 2x + y = 5 –2x + y = 1 2x + y = 5 2(1) + 3 5 2 + 3 5 5 The ordered pair (1, 3) makes both equations true. Substitute 1 for x and 3 for y. –2x + y = 1 –2(1) + 3 1 –2 + 3 1 1 (1, 3) is the solution of the system. Identifying Systems of Solutions

Lesson Quiz: Part I Tell whether the ordered pair is a solution of the given system. 1. (–3, 1); 2. (2, –4); yes no

Lesson Quiz: Part II Solve the system by graphing. 3. 4. Joy has 5 collectable stamps and will buy 2 more each month. Ronald has 25 collectable stamps and will sell 3 each month. After how many months will they have the same number of stamps? How many will that be? (2, 5) 4 months y + 2x = 9 y = 4x – 3 13 stamps

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