## 17Calculus Precalculus - Cramer's Rule

##### 17Calculus

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.

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.

$$x=2, y=-5$$

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

$$x=2, y=-5$$

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$$5x+7y=-1;$$ $$6x+8y=1$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$5x+7y=-1;$$ $$6x+8y=1$$

$$x=15/2, y=11/2$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$5x+7y=-1;$$ $$6x+8y=1$$

Solution

### Thinkwell - 1885 video solution

video by Thinkwell

$$x=15/2, y=11/2$$

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$$4x-2y=10;$$ $$3x-5y=11$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$4x-2y=10;$$ $$3x-5y=11$$

$$x=2, y=-1$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$4x-2y=10;$$ $$3x-5y=11$$

Solution

### mattemath - 1894 video solution

video by mattemath

$$x=2, y=-1$$

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$$x-y=-3;$$ $$x+4y=17$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$x-y=-3;$$ $$x+4y=17$$

$$x=1, y=4$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$x-y=-3;$$ $$x+4y=17$$

Solution

### MIP4U - 1895 video solution

video by MIP4U

$$x=1, y=4$$

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$$2x-3y=16;$$ $$x+2y=1$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x-3y=16;$$ $$x+2y=1$$

$$x=5, y=-2$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x-3y=16;$$ $$x+2y=1$$

Solution

### MIP4U - 1897 video solution

video by MIP4U

$$x=5, y=-2$$

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$$2x + 5y = 26$$
$$5x - 4y = -1$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x + 5y = 26$$
$$5x - 4y = -1$$

$$(3, 4)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x + 5y = 26$$
$$5x - 4y = -1$$

Solution

### 2855 video solution

$$(3, 4)$$

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$$3x - 2y = -4$$
$$4x - y = 3$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$3x - 2y = -4$$
$$4x - y = 3$$

$$(2, 5)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$3x - 2y = -4$$
$$4x - y = 3$$

Solution

### 2856 video solution

$$(2, 5)$$

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$$2x + 3y = 13$$
$$3x - 5y = -9$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x + 3y = 13$$
$$3x - 5y = -9$$

$$(2, 3)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x + 3y = 13$$
$$3x - 5y = -9$$

Solution

### 2857 video solution

$$(2, 3)$$

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$$2x - 7y = 1$$
$$3x + y = 13$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x - 7y = 1$$
$$3x + y = 13$$

$$(4, 1)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x - 7y = 1$$
$$3x + y = 13$$

Solution

### 2858 video solution

$$(4, 1)$$

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$$x - 2y = -3$$
$$3x + y = 5$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$x - 2y = -3$$
$$3x + y = 5$$

$$(1, 2)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$x - 2y = -3$$
$$3x + y = 5$$

Solution

### 2859 video solution

$$(1, 2)$$

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$$x - y = 4$$
$$2x + y = 2$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$x - y = 4$$
$$2x + y = 2$$

$$(2, -2)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$x - y = 4$$
$$2x + y = 2$$

Solution

### 2860 video solution

$$(2, -2)$$

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$$4x - 2y = 10$$
$$3x - 5y = 11$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$4x - 2y = 10$$
$$3x - 5y = 11$$

$$(2, -1)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$4x - 2y = 10$$
$$3x - 5y = 11$$

Solution

### mattemath - 2861 video solution

video by mattemath

$$(2, -1)$$

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$$2x - 3y = 1$$
$$x + 2y = 11$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x - 3y = 1$$
$$x + 2y = 11$$

$$(5, 3)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x - 3y = 1$$
$$x + 2y = 11$$

Solution

### 2862 video solution

$$(5, 3)$$

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$$3x = 2y + 5$$
$$4y = 6x - 8$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$3x = 2y + 5$$
$$4y = 6x - 8$$

no solution

Problem Statement

Solve this linear system using Cramer's Rule.
$$3x = 2y + 5$$
$$4y = 6x - 8$$

Solution

### 2863 video solution

no solution

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$$x + 2y = -3$$
$$-3x - 6y = 9$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$x + 2y = -3$$
$$-3x - 6y = 9$$

infinite number of solutions

Problem Statement

Solve this linear system using Cramer's Rule.
$$x + 2y = -3$$
$$-3x - 6y = 9$$

Solution

### 2864 video solution

infinite number of solutions

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$$3x + 7y = 5$$
$$-2x + y = -9$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$3x + 7y = 5$$
$$-2x + y = -9$$

$$(4, -1)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$3x + 7y = 5$$
$$-2x + y = -9$$

Solution

### 2865 video solution

$$(4, -1)$$

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$$2x + 3y = -4$$
$$-x + y = 7$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x + 3y = -4$$
$$-x + y = 7$$

$$(-5, 2)$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$2x + 3y = -4$$
$$-x + y = 7$$

Solution

### 2866 video solution

$$(-5, 2)$$

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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.
$$2x+3y+z=2;$$ $$-x+2y+3z=-1;$$ $$-3x-3y+z=0$$

$$x=4, y=-3, z=3$$

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

$$x=4, y=-3, z=3$$

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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.
$$2x-y-z=2;$$ $$4x+y-z=-5;$$ $$6x-2y+2z=17$$

$$x=1/2, y=-4, z=3$$

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

$$x=1/2, y=-4, z=3$$

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$$x+2z=9;$$ $$2y+z=8;$$ $$4x-3y=-2$$

Problem Statement

Solve this linear system using Cramer's Rule.
$$x+2z=9;$$ $$2y+z=8;$$ $$4x-3y=-2$$

$$x=1, y=2, z=4$$

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

$$x=1, y=2, z=4$$

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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.
$$x-y+z=4;$$ $$2x+y+z=7;$$ $$-x-2y+2z=-1$$

$$x=3, y=0, z=1$$

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

$$x=3, y=0, z=1$$

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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.
$$-x+2y-3z=1;$$ $$2x+z=0;$$ $$3x-4y+4z=2$$

$$x=-4/5, y=-3/2, z=-8/5$$

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

$$x=-4/5, y=-3/2, z=-8/5$$

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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.
$$x - 3y + 3z = -6$$
$$-2x + 4y + z = 3$$
$$3x - 5y +4z = -9$$

$$(0, 1, -1)$$

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

$$(0, 1, -1)$$

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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.
$$3x - 2y + z = 2$$
$$4x + 3y - 2z = 4$$
$$5x - 3y + 3z = 8$$

$$(1, 2, 3)$$

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

$$(1, 2, 3)$$

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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.
$$x + y - z = -2$$
$$2x - y + z = 0$$
$$x - 2y + 3z = 1$$

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

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

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

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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.
$$2x + y - z = 1$$
$$3x + 2y + 2z = 13$$
$$4x - 2y + 3z = 9$$

$$(1, 2, 3)$$

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

$$(1, 2, 3)$$

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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.
$$2x + 3y - 5z = 1$$
$$x + y - z = 2$$
$$2y + z = 8$$

$$(1, 3, 2)$$

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

$$(1, 3, 2)$$

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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.
$$3x + y - z = 0$$
$$-2x + 5y + 4z = -1$$
$$3x + 2y + z = 1$$

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

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

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

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