17Calculus Precalculus - Cramer's Rule
17Calculus
This page covers how to solve linear systems of equations using determinants.
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We can use matrices and determinants to solve systems of linear equations using a technique called Cramers Rule. We will show how to do this with 2 equations and 2 unknowns. However, the concept can be extended to higher order systems.
Problem
We are given a system of linear equations of the form
\(
\begin{array}{ccccc}
ax & + & by & = & z_0 \\
cx & + & dy & = & z_1
\end{array}
\)
where \( a, b, c, d, z_0, z_1 \) are all real constants and the variables are \( x\) and \(y \).
We need to find what values of \(x\) and \(y\) solve this equation. There are three possible cases.
1. \(x\) and \(y\) are real, unique and not equal.
2. \(x\) and \(y\) are real and equal.
3. \(x\) and \(y\) are complex.
The key to determining which case holds is to look at the determinant of the coefficient matrix, i.e.
\(
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
\)
If this determinant is zero, then we cannot use this technique and either case 2 or 3 hold. If this case is non-zero, then case 1 holds and we can solve this problem. Let's call the coefficient matrix \(A\) and so
\(
\abs{A} =
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
\)
Solution
As long as the determinant of the coefficient matrix is NOT zero, we can solve this system of equations and the values are given by
\(\displaystyle{
x = \frac{
\begin{vmatrix}
z_0 & b \\
z_1 & d
\end{vmatrix}
}{\abs{A}}
}\)
and
\(\displaystyle{
y = \frac{
\begin{vmatrix}
a & z_0 \\
c & z_1
\end{vmatrix}
}{\abs{A}}
}\)
In each case above, notice that we have replaced the column of matrix \(A\) corresponding to the variable we are calculating with the \(z\) constants.
Here are a couple of videos with examples.
PatrickJMT - video 1
video by PatrickJMT |
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PatrickJMT - video 2
video by PatrickJMT |
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The best way to learn Cramer's Rule is by watching someone work specific problems and then working plenty on your own. To get started, here is a video showing, in general, how to use Cramer's Rule, then he does a specific example. After this video, you should be able to work problems on your own.
Thinkwell - Using Cramer's Rule [7min-33secs]
video by Thinkwell |
|---|
Here is a video showing a proof or justification for Cramer's Rule.
PatrickJMT - Cramer's Rule : A Proof / Justification for a System of 2 Linear Equations, 2 Unknowns [10min-12secs]
video by PatrickJMT |
|---|
Important!
Although most of the solutions below do not show the work, it is important to check your work when using Cramer's Rule. As you saw from the videos and you will see in the solutions, it is very, very easy to make mistakes that propagate and give you all wrong answers. So it is important to plug your answers back into all of the equations to make sure you have the correct answers. Your work may be considered incomplete if you do not check your answers (depending on what your instructor expects). So just get used to always check your answers. It could be the difference between a whole letter grade and it takes only a few seconds.
Okay, now you are ready for some practice problems.
Practice - 2x2 Systems
Unless otherwise instructed, solve these linear systems using Cramer's Rule.
\( 3x+4y=-14; \) \( -2x-3y=11 \)
Problem Statement |
|---|
Solve \( 3x+4y=-14; \) \( -2x-3y=11 \) using Cramer's Rule.
Final Answer |
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\(x=2, y=-5\)
Problem Statement
Solve \( 3x+4y=-14; \) \( -2x-3y=11 \) using Cramer's Rule.
Solution
Krista King Math - 1884 video solution
video by Krista King Math |
|---|
Final Answer
\(x=2, y=-5\)
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|---|
\( 5x+7y=-1; \) \( 6x+8y=1 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 5x+7y=-1; \) \( 6x+8y=1 \)
Final Answer |
|---|
\(x=15/2, y=11/2\)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 5x+7y=-1; \) \( 6x+8y=1 \)
Solution
Thinkwell - 1885 video solution
video by Thinkwell |
|---|
Final Answer
\(x=15/2, y=11/2\)
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|---|
\( 4x-2y=10; \) \( 3x-5y=11 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 4x-2y=10; \) \( 3x-5y=11 \)
Final Answer |
|---|
\( x=2, y=-1 \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 4x-2y=10; \) \( 3x-5y=11 \)
Solution
mattemath - 1894 video solution
video by mattemath |
|---|
Final Answer
\( x=2, y=-1 \)
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|---|
\( x-y=-3; \) \( x+4y=17 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( x-y=-3; \) \( x+4y=17 \)
Final Answer |
|---|
\( x=1, y=4 \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( x-y=-3; \) \( x+4y=17 \)
Solution
MIP4U - 1895 video solution
video by MIP4U |
|---|
Final Answer
\( x=1, y=4 \)
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\( 2x-3y=16;\) \( x+2y=1 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x-3y=16;\) \( x+2y=1 \)
Final Answer |
|---|
\( x=5, y=-2 \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x-3y=16;\) \( x+2y=1 \)
Solution
MIP4U - 1897 video solution
video by MIP4U |
|---|
Final Answer
\( x=5, y=-2 \)
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|---|
\( 2x + 5y = 26 \)
\( 5x - 4y = -1 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x + 5y = 26 \)
\( 5x - 4y = -1 \)
Final Answer |
|---|
\( (3, 4) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x + 5y = 26 \)
\( 5x - 4y = -1 \)
Solution
2855 video solution
Final Answer
\( (3, 4) \)
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|---|
\( 3x - 2y = -4 \)
\( 4x - y = 3 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 3x - 2y = -4 \)
\( 4x - y = 3 \)
Final Answer |
|---|
\( (2, 5) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 3x - 2y = -4 \)
\( 4x - y = 3 \)
Solution
2856 video solution
Final Answer
\( (2, 5) \)
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|---|
\( 2x + 3y = 13 \)
\( 3x - 5y = -9 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x + 3y = 13 \)
\( 3x - 5y = -9 \)
Final Answer |
|---|
\( (2, 3) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x + 3y = 13 \)
\( 3x - 5y = -9 \)
Solution
2857 video solution
Final Answer
\( (2, 3) \)
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|---|
\( 2x - 7y = 1 \)
\( 3x + y = 13 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x - 7y = 1 \)
\( 3x + y = 13 \)
Final Answer |
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\( (4, 1) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x - 7y = 1 \)
\( 3x + y = 13 \)
Solution
2858 video solution
Final Answer
\( (4, 1) \)
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|---|
\( x - 2y = -3 \)
\( 3x + y = 5 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( x - 2y = -3 \)
\( 3x + y = 5 \)
Final Answer |
|---|
\( (1, 2) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( x - 2y = -3 \)
\( 3x + y = 5 \)
Solution
2859 video solution
Final Answer
\( (1, 2) \)
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|---|
\( x - y = 4 \)
\( 2x + y = 2 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( x - y = 4 \)
\( 2x + y = 2 \)
Final Answer |
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\( (2, -2) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( x - y = 4 \)
\( 2x + y = 2 \)
Solution
2860 video solution
Final Answer
\( (2, -2) \)
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|---|
\( 4x - 2y = 10 \)
\( 3x - 5y = 11 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 4x - 2y = 10 \)
\( 3x - 5y = 11 \)
Final Answer |
|---|
\( (2, -1) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 4x - 2y = 10 \)
\( 3x - 5y = 11 \)
Solution
mattemath - 2861 video solution
video by mattemath |
|---|
Final Answer
\( (2, -1) \)
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\( 2x - 3y = 1 \)
\( x + 2y = 11 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x - 3y = 1 \)
\( x + 2y = 11 \)
Final Answer |
|---|
\( (5, 3) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x - 3y = 1 \)
\( x + 2y = 11 \)
Solution
2862 video solution
Final Answer
\( (5, 3) \)
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|---|
\( 3x = 2y + 5 \)
\( 4y = 6x - 8 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 3x = 2y + 5 \)
\( 4y = 6x - 8 \)
Final Answer |
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no solution
Problem Statement
Solve this linear system using Cramer's Rule.
\( 3x = 2y + 5 \)
\( 4y = 6x - 8 \)
Solution
2863 video solution
Final Answer
no solution
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|---|
\( x + 2y = -3 \)
\( -3x - 6y = 9 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( x + 2y = -3 \)
\( -3x - 6y = 9 \)
Final Answer |
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infinite number of solutions
Problem Statement
Solve this linear system using Cramer's Rule.
\( x + 2y = -3 \)
\( -3x - 6y = 9 \)
Solution
2864 video solution
Final Answer
infinite number of solutions
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|---|
\( 3x + 7y = 5 \)
\( -2x + y = -9 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 3x + 7y = 5 \)
\( -2x + y = -9 \)
Final Answer |
|---|
\( (4, -1) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 3x + 7y = 5 \)
\( -2x + y = -9 \)
Solution
2865 video solution
Final Answer
\( (4, -1) \)
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|---|
\( 2x + 3y = -4 \)
\( -x + y = 7 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x + 3y = -4 \)
\( -x + y = 7 \)
Final Answer |
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\( (-5, 2) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x + 3y = -4 \)
\( -x + y = 7 \)
Solution
2866 video solution
Final Answer
\( (-5, 2) \)
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Practice - 3x3 Systems
\( 2x+3y+z=2; \) \( -x+2y+3z=-1; \) \( -3x-3y+z=0 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x+3y+z=2; \) \( -x+2y+3z=-1; \) \( -3x-3y+z=0 \)
Final Answer |
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\( x=4, y=-3, z=3 \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x+3y+z=2; \) \( -x+2y+3z=-1; \) \( -3x-3y+z=0 \)
Solution
MIP4U - 1896 video solution
video by MIP4U |
|---|
Final Answer
\( x=4, y=-3, z=3 \)
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\( 2x-y-z=2; \) \( 4x+y-z=-5; \) \( 6x-2y+2z=17 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x-y-z=2; \) \( 4x+y-z=-5; \) \( 6x-2y+2z=17 \)
Final Answer |
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\( x=1/2, y=-4, z=3 \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x-y-z=2; \) \( 4x+y-z=-5; \) \( 6x-2y+2z=17 \)
Solution
MIP4U - 1902 video solution
video by MIP4U |
|---|
Final Answer
\( x=1/2, y=-4, z=3 \)
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\( x+2z=9; \) \( 2y+z=8; \) \( 4x-3y=-2 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( x+2z=9; \) \( 2y+z=8; \) \( 4x-3y=-2 \)
Final Answer |
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\( x=1, y=2, z=4 \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( x+2z=9; \) \( 2y+z=8; \) \( 4x-3y=-2 \)
Solution
PatrickJMT - 1903 video solution
video by PatrickJMT |
|---|
Final Answer
\( x=1, y=2, z=4 \)
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\( x-y+z=4; \) \( 2x+y+z=7; \) \( -x-2y+2z=-1 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( x-y+z=4; \) \( 2x+y+z=7; \) \( -x-2y+2z=-1 \)
Final Answer |
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\( x=3, y=0, z=1 \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( x-y+z=4; \) \( 2x+y+z=7; \) \( -x-2y+2z=-1 \)
Solution
PatrickJMT - 1904 video solution
video by PatrickJMT |
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Final Answer
\( x=3, y=0, z=1 \)
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\( -x+2y-3z=1; \) \( 2x+z=0; \) \( 3x-4y+4z=2 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( -x+2y-3z=1; \) \( 2x+z=0; \) \( 3x-4y+4z=2 \)
Final Answer |
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\( x=-4/5, y=-3/2, z=-8/5 \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( -x+2y-3z=1; \) \( 2x+z=0; \) \( 3x-4y+4z=2 \)
Solution
mattemath - 1905 video solution
video by mattemath |
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Final Answer
\( x=-4/5, y=-3/2, z=-8/5 \)
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\( x - 3y + 3z = -6 \)
\( -2x + 4y + z = 3 \)
\( 3x - 5y +4z = -9 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( x - 3y + 3z = -6 \)
\( -2x + 4y + z = 3 \)
\( 3x - 5y +4z = -9 \)
Final Answer |
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\( (0, 1, -1) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( x - 3y + 3z = -6 \)
\( -2x + 4y + z = 3 \)
\( 3x - 5y +4z = -9 \)
Solution
This problem is solved in 2 consecutive videos.
2867 video solution
2867 video solution
Final Answer
\( (0, 1, -1) \)
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\( 3x - 2y + z = 2 \)
\( 4x + 3y - 2z = 4 \)
\( 5x - 3y + 3z = 8 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 3x - 2y + z = 2 \)
\( 4x + 3y - 2z = 4 \)
\( 5x - 3y + 3z = 8 \)
Final Answer |
|---|
\( (1, 2, 3) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 3x - 2y + z = 2 \)
\( 4x + 3y - 2z = 4 \)
\( 5x - 3y + 3z = 8 \)
Solution
2868 video solution
Final Answer
\( (1, 2, 3) \)
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\( x + y - z = -2 \)
\( 2x - y + z = 0 \)
\( x - 2y + 3z = 1 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( x + y - z = -2 \)
\( 2x - y + z = 0 \)
\( x - 2y + 3z = 1 \)
Final Answer |
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\( (-2/3, -7/3, -1) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( x + y - z = -2 \)
\( 2x - y + z = 0 \)
\( x - 2y + 3z = 1 \)
Solution
2870 video solution
Final Answer
\( (-2/3, -7/3, -1) \)
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\( 2x + y - z = 1 \)
\( 3x + 2y + 2z = 13 \)
\( 4x - 2y + 3z = 9 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x + y - z = 1 \)
\( 3x + 2y + 2z = 13 \)
\( 4x - 2y + 3z = 9 \)
Final Answer |
|---|
\( (1, 2, 3) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x + y - z = 1 \)
\( 3x + 2y + 2z = 13 \)
\( 4x - 2y + 3z = 9 \)
Solution
2871 video solution
Final Answer
\( (1, 2, 3) \)
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\( 2x + 3y - 5z = 1 \)
\( x + y - z = 2 \)
\( 2y + z = 8 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 2x + 3y - 5z = 1 \)
\( x + y - z = 2 \)
\( 2y + z = 8 \)
Final Answer |
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\( (1, 3, 2) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 2x + 3y - 5z = 1 \)
\( x + y - z = 2 \)
\( 2y + z = 8 \)
Solution
2872 video solution
Final Answer
\( (1, 3, 2) \)
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\( 3x + y - z = 0 \)
\( -2x + 5y + 4z = -1 \)
\( 3x + 2y + z = 1 \)
Problem Statement |
|---|
Solve this linear system using Cramer's Rule.
\( 3x + y - z = 0 \)
\( -2x + 5y + 4z = -1 \)
\( 3x + 2y + z = 1 \)
Final Answer |
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\( (1/2, -2/3, 5/6) \)
Problem Statement
Solve this linear system using Cramer's Rule.
\( 3x + y - z = 0 \)
\( -2x + 5y + 4z = -1 \)
\( 3x + 2y + z = 1 \)
Solution
2873 video solution
Final Answer
\( (1/2, -2/3, 5/6) \)
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