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17Calculus Precalculus - Cramer's Rule

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This page covers how to solve linear systems of equations using determinants.

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.

How to Develop a Brilliant Memory Week by Week: 50 Proven Ways to Enhance Your Memory Skills

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

\(x=2, y=-5\)

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

\( (4, 1) \)

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

\( (2, -2) \)

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

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

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

\( (-5, 2) \)

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

\( x=4, y=-3, z=3 \)

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

\( x=1/2, y=-4, z=3 \)

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

\( x=1, y=2, z=4 \)

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

\( x=3, y=0, z=1 \)

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

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

\( x=-4/5, y=-3/2, z=-8/5 \)

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

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

\( (0, 1, -1) \)

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

\( (-2/3, -7/3, -1) \)

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

\( (1, 3, 2) \)

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

\( (1/2, -2/3, 5/6) \)

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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Topics You Need To Understand For This Page

basics of matrices

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Unless otherwise instructed, solve these linear systems using Cramer's Rule.

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