17Calculus Precalculus - Cramer's Rule
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 |
|---|
PatrickJMT - video 2
video by PatrickJMT |
|---|
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
Solve these linear systems using Cramer's Rule.
Let's start with some 2x2 systems
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x+4y=-14; \) \( -2x-3y=11 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x+4y=-14; \) \( -2x-3y=11 \)
Final Answer |
|---|
\(x=2, y=-5\)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x+4y=-14; \) \( -2x-3y=11 \)
Solution |
|---|
1884 video
video by Krista King Math |
|---|
Final Answer |
|---|
\(x=2, y=-5\) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 5x+7y=-1; \) \( 6x+8y=1 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 5x+7y=-1; \) \( 6x+8y=1 \)
Final Answer |
|---|
\(x=15/2, y=11/2\)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 5x+7y=-1; \) \( 6x+8y=1 \)
Solution |
|---|
1885 video
video by Thinkwell |
|---|
Final Answer |
|---|
\(x=15/2, y=11/2\) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x-2y=10; \) \( 3x-5y=11 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x-2y=10; \) \( 3x-5y=11 \)
Final Answer |
|---|
\( x=2, y=-1 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x-2y=10; \) \( 3x-5y=11 \)
Solution |
|---|
1894 video
video by mattemath |
|---|
Final Answer |
|---|
\( x=2, y=-1 \) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x-y=-3; \) \( x+4y=17 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x-y=-3; \) \( x+4y=17 \)
Final Answer |
|---|
\( x=1, y=4 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x-y=-3; \) \( x+4y=17 \)
Solution |
|---|
1895 video
video by MIP4U |
|---|
Final Answer |
|---|
\( x=1, y=4 \) |
|
close solution
|
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|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x-3y=16;\) \( x+2y=1 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x-3y=16;\) \( x+2y=1 \)
Final Answer |
|---|
\( x=5, y=-2 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x-3y=16;\) \( x+2y=1 \)
Solution |
|---|
1897 video
video by MIP4U |
|---|
Final Answer |
|---|
\( x=5, y=-2 \) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 5y = 26 \)
\( 5x - 4y = -1 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 5y = 26 \)
\( 5x - 4y = -1 \)
Final Answer |
|---|
\( (3, 4) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 5y = 26 \)
\( 5x - 4y = -1 \)
Solution |
|---|
2855 video
Final Answer |
|---|
\( (3, 4) \) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x - 2y = -4 \)
\( 4x - y = 3 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x - 2y = -4 \)
\( 4x - y = 3 \)
Final Answer |
|---|
\( (2, 5) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x - 2y = -4 \)
\( 4x - y = 3 \)
Solution |
|---|
2856 video
Final Answer |
|---|
\( (2, 5) \) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = 13 \)
\( 3x - 5y = -9 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = 13 \)
\( 3x - 5y = -9 \)
Final Answer |
|---|
\( (2, 3) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = 13 \)
\( 3x - 5y = -9 \)
Solution |
|---|
2857 video
Final Answer |
|---|
\( (2, 3) \) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x - 7y = 1 \)
\( 3x + y = 13 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x - 7y = 1 \)
\( 3x + y = 13 \)
Final Answer |
|---|
\( (4, 1) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x - 7y = 1 \)
\( 3x + y = 13 \)
Solution |
|---|
2858 video
Final Answer |
|---|
\( (4, 1) \) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x - 2y = -3 \)
\( 3x + y = 5 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x - 2y = -3 \)
\( 3x + y = 5 \)
Final Answer |
|---|
\( (1, 2) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x - 2y = -3 \)
\( 3x + y = 5 \)
Solution |
|---|
2859 video
Final Answer |
|---|
\( (1, 2) \) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x - y = 4 \)
\( 2x + y = 2 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x - y = 4 \)
\( 2x + y = 2 \)
Final Answer |
|---|
\( (2, -2) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x - y = 4 \)
\( 2x + y = 2 \)
Solution |
|---|
2860 video
Final Answer |
|---|
\( (2, -2) \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x - 2y = 10 \)
\( 3x - 5y = 11 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x - 2y = 10 \)
\( 3x - 5y = 11 \)
Final Answer |
|---|
\( (2, -1) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 4x - 2y = 10 \)
\( 3x - 5y = 11 \)
Solution |
|---|
2861 video
video by mattemath |
|---|
Final Answer |
|---|
\( (2, -1) \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x - 3y = 1 \)
\( x + 2y = 11 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x - 3y = 1 \)
\( x + 2y = 11 \)
Final Answer |
|---|
\( (5, 3) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x - 3y = 1 \)
\( x + 2y = 11 \)
Solution |
|---|
2862 video
Final Answer |
|---|
\( (5, 3) \) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x = 2y + 5 \)
\( 4y = 6x - 8 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x = 2y + 5 \)
\( 4y = 6x - 8 \)
Final Answer |
|---|
no solution
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x = 2y + 5 \)
\( 4y = 6x - 8 \)
Solution |
|---|
2863 video
Final Answer |
|---|
no solution |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + 2y = -3 \)
\( -3x - 6y = 9 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + 2y = -3 \)
\( -3x - 6y = 9 \)
Final Answer |
|---|
infinite number of solutions
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + 2y = -3 \)
\( -3x - 6y = 9 \)
Solution |
|---|
2864 video
Final Answer |
|---|
infinite number of solutions |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + 7y = 5 \)
\( -2x + y = -9 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + 7y = 5 \)
\( -2x + y = -9 \)
Final Answer |
|---|
\( (4, -1) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + 7y = 5 \)
\( -2x + y = -9 \)
Solution |
|---|
2865 video
Final Answer |
|---|
\( (4, -1) \) |
|
close solution
|
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|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = -4 \)
\( -x + y = 7 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = -4 \)
\( -x + y = 7 \)
Final Answer |
|---|
\( (-5, 2) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y = -4 \)
\( -x + y = 7 \)
Solution |
|---|
2866 video
Final Answer |
|---|
\( (-5, 2) \) |
|
close solution
|
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Now you are ready for some 3x3 systems.
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x+3y+z=2; \) \( -x+2y+3z=-1; \) \( -3x-3y+z=0 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x+3y+z=2; \) \( -x+2y+3z=-1; \) \( -3x-3y+z=0 \)
Final Answer |
|---|
\( x=4, y=-3, z=3 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x+3y+z=2; \) \( -x+2y+3z=-1; \) \( -3x-3y+z=0 \)
Solution |
|---|
1896 video
video by MIP4U |
|---|
Final Answer |
|---|
\( x=4, y=-3, z=3 \) |
|
close solution
|
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|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x-y-z=2; \) \( 4x+y-z=-5; \) \( 6x-2y+2z=17 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x-y-z=2; \) \( 4x+y-z=-5; \) \( 6x-2y+2z=17 \)
Final Answer |
|---|
\( x=1/2, y=-4, z=3 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x-y-z=2; \) \( 4x+y-z=-5; \) \( 6x-2y+2z=17 \)
Solution |
|---|
1902 video
video by MIP4U |
|---|
Final Answer |
|---|
\( x=1/2, y=-4, z=3 \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x+2z=9; \) \( 2y+z=8; \) \( 4x-3y=-2 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x+2z=9; \) \( 2y+z=8; \) \( 4x-3y=-2 \)
Final Answer |
|---|
\( x=1, y=2, z=4 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x+2z=9; \) \( 2y+z=8; \) \( 4x-3y=-2 \)
Solution |
|---|
1903 video
video by PatrickJMT |
|---|
Final Answer |
|---|
\( x=1, y=2, z=4 \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x-y+z=4; \) \( 2x+y+z=7; \) \( -x-2y+2z=-1 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x-y+z=4; \) \( 2x+y+z=7; \) \( -x-2y+2z=-1 \)
Final Answer |
|---|
\( x=3, y=0, z=1 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x-y+z=4; \) \( 2x+y+z=7; \) \( -x-2y+2z=-1 \)
Solution |
|---|
1904 video
video by PatrickJMT |
|---|
Final Answer |
|---|
\( x=3, y=0, z=1 \) |
|
close solution
|
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( -x+2y-3z=1; \) \( 2x+z=0; \) \( 3x-4y+4z=2 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( -x+2y-3z=1; \) \( 2x+z=0; \) \( 3x-4y+4z=2 \)
Final Answer |
|---|
\( x=-4/5, y=-3/2, z=-8/5 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( -x+2y-3z=1; \) \( 2x+z=0; \) \( 3x-4y+4z=2 \)
Solution |
|---|
1905 video
video by mattemath |
|---|
Final Answer |
|---|
\( x=-4/5, y=-3/2, z=-8/5 \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x - 3y + 3z = -6 \)
\( -2x + 4y + z = 3 \)
\( 3x - 5y +4z = -9 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x - 3y + 3z = -6 \)
\( -2x + 4y + z = 3 \)
\( 3x - 5y +4z = -9 \)
Final Answer |
|---|
\( (0, 1, -1) \)
Problem Statement |
|---|
Unless otherwise instructed, 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
2867 video
Final Answer |
|---|
\( (0, 1, -1) \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
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Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x - 2y + z = 2 \)
\( 4x + 3y - 2z = 4 \)
\( 5x - 3y + 3z = 8 \)
Problem Statement |
|---|
Unless otherwise instructed, 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 |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x - 2y + z = 2 \)
\( 4x + 3y - 2z = 4 \)
\( 5x - 3y + 3z = 8 \)
Solution |
|---|
2868 video
Final Answer |
|---|
\( (1, 2, 3) \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + y - z = -2 \)
\( 2x - y + z = 0 \)
\( x - 2y + 3z = 1 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + y - z = -2 \)
\( 2x - y + z = 0 \)
\( x - 2y + 3z = 1 \)
Final Answer |
|---|
\( (-2/3, -7/3, -1) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( x + y - z = -2 \)
\( 2x - y + z = 0 \)
\( x - 2y + 3z = 1 \)
Solution |
|---|
2870 video
Final Answer |
|---|
\( (-2/3, -7/3, -1) \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + y - z = 1 \)
\( 3x + 2y + 2z = 13 \)
\( 4x - 2y + 3z = 9 \)
Problem Statement |
|---|
Unless otherwise instructed, 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 |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + y - z = 1 \)
\( 3x + 2y + 2z = 13 \)
\( 4x - 2y + 3z = 9 \)
Solution |
|---|
2871 video
Final Answer |
|---|
\( (1, 2, 3) \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y - 5z = 1 \)
\( x + y - z = 2 \)
\( 2y + z = 8 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y - 5z = 1 \)
\( x + y - z = 2 \)
\( 2y + z = 8 \)
Final Answer |
|---|
\( (1, 3, 2) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 2x + 3y - 5z = 1 \)
\( x + y - z = 2 \)
\( 2y + z = 8 \)
Solution |
|---|
2872 video
Final Answer |
|---|
\( (1, 3, 2) \) |
|
close solution
|
Log in to rate this practice problem and to see it's current rating. |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + y - z = 0 \)
\( -2x + 5y + 4z = -1 \)
\( 3x + 2y + z = 1 \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + y - z = 0 \)
\( -2x + 5y + 4z = -1 \)
\( 3x + 2y + z = 1 \)
Final Answer |
|---|
\( (1/2, -2/3, 5/6) \)
Problem Statement |
|---|
Unless otherwise instructed, solve this linear system using Cramer's Rule.
\( 3x + y - z = 0 \)
\( -2x + 5y + 4z = -1 \)
\( 3x + 2y + z = 1 \)
Solution |
|---|
2873 video
Final Answer |
|---|
\( (1/2, -2/3, 5/6) \) |
|
close solution
|
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|---|
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Help Keep 17Calculus Free |
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The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free. |
