Finite Math Examples

Solve Using a Matrix with Cramer's Rule y=4x+3x-2 , y=5x
y=4x+3x-2 , y=5x
Step 1
Move all of the variables to the left side of each equation.
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Step 1.1
Move all terms containing variables to the left side of the equation.
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Step 1.1.1
Subtract 4x from both sides of the equation.
y-4x=3x-2
y=5x
Step 1.1.2
Subtract 3x from both sides of the equation.
y-4x-3x=-2
y=5x
y-4x-3x=-2
y=5x
Step 1.2
Subtract 3x from -4x.
y-7x=-2
y=5x
Step 1.3
Reorder y and -7x.
-7x+y=-2
y=5x
Step 1.4
Subtract 5x from both sides of the equation.
-7x+y=-2
y-5x=0
Step 1.5
Reorder y and -5x.
-7x+y=-2
-5x+y=0
-7x+y=-2
-5x+y=0
Step 2
Represent the system of equations in matrix format.
[-71-51][xy]=[-20]
Step 3
Find the determinant of the coefficient matrix [-71-51].
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Step 3.1
Write [-71-51] in determinant notation.
|-71-51|
Step 3.2
The determinant of a 2×2 matrix can be found using the formula |abcd|=ad-cb.
-71-(-51)
Step 3.3
Simplify the determinant.
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Step 3.3.1
Simplify each term.
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Step 3.3.1.1
Multiply -7 by 1.
-7-(-51)
Step 3.3.1.2
Multiply -(-51).
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Step 3.3.1.2.1
Multiply -5 by 1.
-7--5
Step 3.3.1.2.2
Multiply -1 by -5.
-7+5
-7+5
-7+5
Step 3.3.2
Add -7 and 5.
-2
-2
D=-2
Step 4
Since the determinant is not 0, the system can be solved using Cramer's Rule.
Step 5
Find the value of x by Cramer's Rule, which states that x=DxD.
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Step 5.1
Replace column 1 of the coefficient matrix that corresponds to the x-coefficients of the system with [-20].
|-2101|
Step 5.2
Find the determinant.
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Step 5.2.1
The determinant of a 2×2 matrix can be found using the formula |abcd|=ad-cb.
-21+01
Step 5.2.2
Simplify the determinant.
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Step 5.2.2.1
Simplify each term.
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Step 5.2.2.1.1
Multiply -2 by 1.
-2+01
Step 5.2.2.1.2
Multiply 0 by 1.
-2+0
-2+0
Step 5.2.2.2
Add -2 and 0.
-2
-2
Dx=-2
Step 5.3
Use the formula to solve for x.
x=DxD
Step 5.4
Substitute -2 for D and -2 for Dx in the formula.
x=-2-2
Step 5.5
Divide -2 by -2.
x=1
x=1
Step 6
Find the value of y by Cramer's Rule, which states that y=DyD.
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Step 6.1
Replace column 2 of the coefficient matrix that corresponds to the y-coefficients of the system with [-20].
|-7-2-50|
Step 6.2
Find the determinant.
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Step 6.2.1
The determinant of a 2×2 matrix can be found using the formula |abcd|=ad-cb.
-70-(-5-2)
Step 6.2.2
Simplify the determinant.
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Step 6.2.2.1
Simplify each term.
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Step 6.2.2.1.1
Multiply -7 by 0.
0-(-5-2)
Step 6.2.2.1.2
Multiply -(-5-2).
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Step 6.2.2.1.2.1
Multiply -5 by -2.
0-110
Step 6.2.2.1.2.2
Multiply -1 by 10.
0-10
0-10
0-10
Step 6.2.2.2
Subtract 10 from 0.
-10
-10
Dy=-10
Step 6.3
Use the formula to solve for y.
y=DyD
Step 6.4
Substitute -2 for D and -10 for Dy in the formula.
y=-10-2
Step 6.5
Divide -10 by -2.
y=5
y=5
Step 7
List the solution to the system of equations.
x=1
y=5
 [x2  12  π  xdx ]