MORE SYSTEMS OF EQUATIONS
First put both equations into standard form (Ax + By = C) |
Elimination Using Addition |
Example #1: Solve the system shown below using the elimination method.
The solution to this system of equations is (3, 4). |
2x + y = 9 |
3x – y = 16 |
If we add to eliminate the y's, what equation will we then have?
5x = 25
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What is the value of x?
x = 5
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If we substitute the value of x in the first equation, what equation will we have?
2(5) + y = 9 or 10 + y = 9
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What is the value of y?
y = –1
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What is the ordered pair that solves this system of equation?
(5, –1)
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How can the answer be checked in the first equation?
2(5) + –1 = 9
10 + –1 = 9
9 = 9
Checked!
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How can the answer be checked in the second equation?
3(5) – (–1) = 16
15 + 1 = 16
16 = 16
Checked!
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Elimination Using Subtraction |
Example #2: Solve the system shown below using the elimination method.
The solution to this system of equations is (1, 1). |
2x – 5y = –6 |
2x + y = 12 |
If we subtract to eliminate the x's, what equation will we then have?
Note: In subtraction, change the signs to their opposites, then add.
–6y = –18
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What is the value of y?
y = 3
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If we substitute the value of y in the first equation, what equation will we have?
2x + 3 = 12
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What is the value of x?
x = 4.5
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What is the ordered pair that solves this system of equation?
(4.5, 3)
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How can the answer be checked in the first equation?
2(4.5) – 5(3) = –6
9 – 15 = –6
–6 = –6
Checked!
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How can the answer be checked in the second equation?
2(4.5) + 3 = 12
9 + 3 = 12
12 = 12
Checked!
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Elimination Using Multiplication and Addition or Subtraction |
Example #3: Solve the system shown below using the elimination method.
The solution to this system of equations is (2, –2). |
Example #4: Solve the system shown below using the elimination method.
The solution to this system of equations (3, –2). |
4x + 3y = –1 |
5x + 4y = 1 |
If we decide to eliminate y's, how will be make the coefficients the same?
Multiply the first equation by 4 and the second equation by 3.
4(4x + 3y = –1) and 3(5x + 4y = 1)
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What equation will we use in place of the first equation?
16x + 12y = –4
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What equation will we use in place of the second equation?
15x + 12y = 3
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If we subtract to eliminate the y's, what is the value of x?
x = –7
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If we substitute the value of x in the first equation, what equation will we have?
4(–7) + 3y = –1 or –28 + 3y = –1
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What is the value of y?
y = 9
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What is the ordered pair that solves this system of equation?
(–7, 9)
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How can the answer be checked in the first equation?
4 (–7) + 3(9) = –1
–28 + 27 = –1
–1= –1
Checked!
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How can the answer be checked in the second equation?
5(–7) + 4(9) = 1
–35 + 36 = 1
1 = 1
Checked!
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To determine if a system is consistent or inconsistent, you need to solve it algebraically, either by substitution or elimination, or graphically. For our purposes, we will solve all systems algebraically.
Example #1: Solve the system of equations shown below.
The solution is (–1, 0), so the system has one unique solution and is consistent. |
Is the point of intersection the same as the solution above?
Yes, (–1, 0)
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Check to see if both equations are graphed correctly.
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Example #2: Solve the system of equations shown below.
Since –4 does not equal 6, there is no ordered pair that satisfies the system; therefore, the system is inconsistent. |
What is true about the graphs of the two equations?
The lines are parallel.
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When solving systems of equations algebraically,
how will we know when the lines are parallel?
The variables will drop out and the final statement will be false.
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Check to see if both equations are graphed correctly.
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Example #3: Solve the system of equations shown below.
Since 0 = 0 for any value of x, the system of equations has infinite solutions. Every ordered pair (x, y) satisfies both equations. The system is consistent (and dependent which is discussed in the next section). The two equations describe the same line. |
What is true about the graphs of the two equations?
The graphs of the lines are the same line.
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When solving systems of equations algebraically, how can it be
determined that there are an infinite number of solutions to the system?
The variables will drop out and the final statement be 0 = 0.
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Check to see if both equations are graphed correctly.
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Example #1: Solve the system of equations shown below.
The solution is ( , –3). Since the system has one solution, this means the system is independent. |
Example #2: Solve the system of equations shown below.
Since this solution produces a true statement that 0 = 0, the solution has many solutions and this means that the system is dependent. |
Example #3: Solve the system of equations shown below.
The solution is (8, 9). Since the system has one solution, this means the system is independent. |
Summary for the Types of Systems of Equations | ||
The solution to a system of equations can be described as follows. Inconsistent Systems will have no solution. The lines of the equations are parallel. Consistent Systems will have one or an infinite number of solutions
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Example #1: The Jets scored 4 more points than the Vets. The total of their scores was 38. How many points did each team score?
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Example #2: Four cans of tuna and 2 boxes of rice cost $7.40. Six cans of tuna and 2 boxes of rice cost $9.70. Find the cost of each item.
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Solving Systems of Equations |
Solving Systems of Equations by Elimination |