1 topic 5.1.1 the graphing method. 2 california standard: 9.0 students solve a system of two linear...
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Topic 5.1.1Topic 5.1.1
The Graphing MethodThe Graphing Method
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The Graphing MethodThe Graphing Method
California Standard: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.
What it means for you:You’ll solve systems of linear equations by graphing the lines and working out where they intersect.
Key Words:• system of linear equations• simultaneous equations
Topic5.1.1
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The Graphing MethodThe Graphing Method
In Section 4.5 you graphed two inequalities to find the region of points that satisfied both inequalities.
Plotting two linear equations on a graph involves fewer steps, and it means you can show the joint solution to the equations graphically.
Topic5.1.1
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The Graphing MethodThe Graphing Method
Systems of Linear Equations
A system of linear equations consists of two or more linear equations in the same variables.
Topic5.1.1
For example: 3x + 2y = 7 and x – 3y = –5 form a
system of linear equations in two variables — x and y.
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The Graphing MethodThe Graphing MethodTopic5.1.1
The solution of a system of linear equations in two variables is a pair of values like x and y, or (x, y), that satisfies each of the equations in the system.
For example, x = 1, y = 2 or (1, 2) is the solution of the
system of equations 3x + 2y = 7 and x – 3y = –5 , since it
satisfies both equations:
3x + 2y = 73(1) + 2(2) = 7
3 + 4 = 7
3x + 2y = 73(1) + 2(2) = 7
3 + 4 = 7
x – 3y = –51 – 3(2) = –5
1 – 6 = –5
x – 3y = –51 – 3(2) = –5
1 – 6 = –5
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The Graphing MethodThe Graphing MethodTopic5.1.1
Equations in a system are often called simultaneous equations because any solution has to satisfy the equations simultaneously (at the same time).
The equations can’t be solved independently of one another.
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Solving Systems of Equations by Graphing
Topic5.1.1
A system of two linear equations can be solved graphically, by graphing both equations in the same coordinate plane.
Every point on the line of an equation is a solution of that equation.
The point at which the two lines cross lies on both lines and so is the solution of both equations.
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The Graphing MethodThe Graphing Method
The solution of a system of linear equations in two variables is the point of intersection (x, y) of their graphs.
Topic5.1.1
Point of intersection
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The Graphing MethodThe Graphing Method
Example 1
Solve this system of equations by graphing:
Solution follows…
Topic5.1.1
2x – 3y = 7
–2x + y = –1
Step 1: Graph both equations in the same coordinate plane.
Solution
Line of first equation: Line of second equation: 2x – 3y = 7
–2x + y = –1
The line goes through the points (2, –1) and (–1, –3).
The line goes through the points (0, –1) and (1, 1).
y = 2x –1
3y = 2x – 7
y = x – 2
3
7
3
x y
2 –1
–1 –3
x y
0 –1
1 1
First find some points that lie on each of the lines.
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Example 1
Solution continues…
Now you can draw the graph:
Solution (continued)
Topic5.1.1
Step 2: Read off the coordinates of the point of intersection.
(–1, –3)
(1, 1)
(2, –1)(0, –1)
y = 2x –1
y = x – 23
73
The point of intersection is (–1, –3). (–1, –3)
Solve this system of equations by graphing: 2x – 3y = 7
–2x + y = –1
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Example 1
Solution (continued)
Topic5.1.1
Step 3: Check whether your coordinates give true statements when they are substituted into each equation.
So x = –1, y = –3 is the solution of the system of equations.
2x – 3y = 7 2(–1) – 3(–3) = 7 7 = 7 — True statement
–2x + y = –1 –2(–1) + (–3) = –1 –1 = –1 — True statement
Solve this system of equations by graphing: 2x – 3y = 7
–2x + y = –1
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Guided Practice
Solution follows…
Solve each system of equations in Exercises 1–2by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
2. y + x = 3 and 3y – x = 51. y + x = 2 and y = x + 22
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y + x = 2
2
3y = x + 2
y + x = 3
3y – x = 5
(0, 2) (1, 2)
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Guided Practice
Solution follows…
3. y = x – 3 and y + 2x = 3
Solve each system of equations in Exercises 3–4by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
4. y – x = 1 and y + x = –33
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2
y + 2x = 3
y = x – 3
y – x = 13
2
y + x = –31
2
(2, –1) (–2, –2)
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Guided Practice
Solution follows…
5. y – x = 3 and y + x = –1
Solve each system of equations in Exercises 5–6by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
6. 2y – x = –6 and y + x = –31
2
y – x = 3
y + x = –1
(–2, 1)
y + x = –31
22y – x = –6
(0, –3)
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Solution follows…
Solve each system of equations in Exercises 1–2by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
2. x + y = 0 and y = –2x1. 2x + y = 7 and y = x + 1
2x + y = 7
y = x + 1
(2, 3)
Independent Practice
x + y = 0
y = –2x
(0, 0)
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The Graphing MethodThe Graphing Method
Solution follows…
Solve each system of equations in Exercises 3–4by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
4. x – y = 4 and x + 4y = –13. y = –3 and x – y = 2
y = –3
x – y = 2
(–1, –3)
Independent Practice
x – y = 4
x + 4y = –1
(3, –1)
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Solution follows…
Solve each system of equations in Exercises 5–6by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
6. y = –x and y = 4x5. 2y + 4x = 4 and y = –x + 3
2y + 4x = 4
y = –x + 3
(–1, 4)
Independent Practice
y = –xy = 4x
(0, 0)
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Solution follows…
Determine the solution to the systems of equations graphed in Exercises 7 and 8.
Topic5.1.1
8. 7.
y = –x
Independent Practice
y = 2x – 7
x + y = –4
y = x – 41
3
(3, –3) (2, –3)
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Solution follows…
Solve each system of equations in Exercises 9–10by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
10. y = 2x – 1 and x + y = 89. x – y = 6 and x + y = 0
x + y = 0
Independent Practice
y = 2x – 1
x + y = 8
x – y = 6
(3, 5)(3, –3)
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Solution follows…
Solve each system of equations in Exercises 11–12by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
12. x – y = 0 and x + y = 811. 4x – 3y = 0 and 4x + y = 16
4x – 3y = 04x + y = 16
(3, 4)
Independent Practice
x – y = 0
x + y = 8
(4, 4)
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Solution follows…
Solve each system of equations in Exercises 13–14by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
14. x – y = 1 and x + y = –313. y = –x + 6 and x – y = –4
y = –x + 6x – y = –4
(1, 5)
Independent Practice
x – y = 1 x + y = –3
(–1, –2)
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Solution follows…
Solve each system of equations in Exercises 15–16by graphing on x- and y-axes spanning from –6 to 6.
Topic5.1.1
16. 2x + y = –8 and 3x + y = –1315. x + y = 1 and x – 2y = 1
x + y = 1
x – 2y = 1
Independent Practice
2x + y = –8
3x + y = –13
(1, 0) (–5, 2)
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Round UpRound Up
There’s something very satisfying about taking two long linear equations and coming up with just a one-coordinate-pair solution.
You should always substitute your solution back into the original equations, to check that you’ve got the correct answer.
The Graphing MethodThe Graphing MethodTopic5.1.1