Using Cramer’s Rule, what is the value of x in the system of linear equations below? 2x-y=3 5x-2y=8

Cramer's rule has you compute 3 determinants, based on different matrices of the coefficients. We can call them A, B, C. Then x and y are found from their ratios.

[tex]A=det\left[\begin{array}{cc}2&-1\\5&-2\end{array}\right] =2(-2)-5(-1)=1\\\\B=det\left[\begin{array}{cc}3&-1\\8&-2\end{array}\right] =3(-2)-8(-1)=2\\\\C=det\left[\begin{array}{cc}2&3\\5&8\end{array}\right] =2(8)-5(3)=1\\\\x=\dfrac{B}{A}=\dfrac{2}{1}=2\\\\y=\dfrac{C}{A}=\dfrac{1}{1}=1\\\\(x,y)=(2,1)[/tex]

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Comment Cramer's Rule

When the solution is some strange fraction, or when you only need one of the variable values (as here), Cramer's rule can get you there pretty directly.

The pattern of coefficient usage in Cramer's rule can be difficult to remember. If you swap the columns of each of the matrices, the pattern can be easier to remember, so that you can even do the math in your head.

psm22415 psm22415 · Answer 2 · 2022-02-18T04:13:46+01:00

By using Cramer’s Rule, the value of x in the system of linear equations is x = 2 and y =1.

Cramer’s Rule;

Cramer’s Rule is defined as the determining method to find the solution of the linear system of equations.

Given

The system of equations;

[tex]\rm 2x-y=3 \\\\5x-2y=8[/tex]

The Cramer's rule to find the value of x in the following system of equations.

[tex]\rm \dfrac{x}{D_1}=\dfrac{y}{D_2}=\dfrac{1}{D}[/tex]

Where;

[tex]\rm D_1=\left[\begin{array}{ccc}3&-1\\8&-2\\\end{array}\right] \\\\D_1=3\times (-2)-(-1)\times 8\\\\D_1=-6-(-8)\\\\D_1=-6+8\\\\D_1=2[/tex]

[tex]\rm D_2=\left[\begin{array}{ccc}2&3\\5&8\\\end{array}\right] \\\\D_2=2\times 8-5\times 3\\\\D_2=16-15\\\\D_2=1[/tex]

[tex]\rm \rm D=\left[\begin{array}{ccc}2&-1\\5&-2\\\end{array}\right] \\\\D=2\times (-2)-(-1)\times 5\\\\D=-4-(-5)\\\\D=-4+5\\\\D=1[/tex]

Therefore,

[tex]\rm \dfrac{x}{2}=\dfrac{y}{1}=\dfrac{1}{1}\\\\\rm \dfrac{x}{2}=\dfrac{1}{1}\\\\x=2\\\\And \ \dfrac{y}{1}=\dfrac{1}{1}\\\\y=1[/tex]

Hence, by using Cramer’s Rule, the value of x in the system of linear equations is x = 2 and y =1.

To know more about Cramer’s Rule click the link given below.

https://brainly.com/question/10132289