Type The Correct Answer In Each Box. Use Numerals Instead Of Words.This System Of Equations Has Been Placed In A Matrix:$\[ \begin{array}{l} y = 650x + 175 \\ y = 25,080 - 120x \end{array} \\]

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Solving a System of Linear Equations using a Matrix

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Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. These equations are linear because they can be written in the form of a linear equation, which is an equation in which the highest power of the variable(s) is 1. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will use a matrix to solve a system of linear equations.

The Matrix Representation of a System of Linear Equations

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of solving a system of linear equations, a matrix can be used to represent the coefficients of the variables and the constants in the equations. The matrix representation of a system of linear equations is given by:

{ \begin{array}{l} y = 650x + 175 \\ y = 25,080 - 120x \end{array} \}

This matrix represents two equations with two variables, x and y. The coefficients of the variables are given by the numbers in the matrix, and the constants are given by the numbers outside the matrix.

Solving the System of Linear Equations using a Matrix

To solve the system of linear equations using a matrix, we need to find the values of x and y that satisfy both equations. We can do this by using the following steps:

  1. Write the matrix equation: Write the matrix equation that represents the system of linear equations. In this case, the matrix equation is:

{ \begin{array}{l} 650x + 175 = y \\ 25,080 - 120x = y \end{array} \}

  1. Set the two equations equal to each other: Set the two equations equal to each other, since they both equal y. This gives us:

650x+175=25,080−120x{ 650x + 175 = 25,080 - 120x }

  1. Solve for x: Solve for x by isolating it on one side of the equation. We can do this by adding 120x to both sides of the equation, which gives us:

770x+175=25,080{ 770x + 175 = 25,080 }

  1. Subtract 175 from both sides: Subtract 175 from both sides of the equation to get:

770x=24,905{ 770x = 24,905 }

  1. Divide both sides by 770: Divide both sides of the equation by 770 to get:

x=32.3{ x = 32.3 }

  1. Find the value of y: Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. We will use the first equation:

y=650x+175{ y = 650x + 175 }

Substituting x = 32.3 into this equation gives us:

y=650(32.3)+175{ y = 650(32.3) + 175 }

y=20,910+175{ y = 20,910 + 175 }

y=21,085{ y = 21,085 }

Conclusion

In this article, we used a matrix to solve a system of linear equations. We first wrote the matrix equation that represents the system of linear equations, then set the two equations equal to each other. We solved for x by isolating it on one side of the equation, and then found the value of y by substituting x into one of the original equations. The values of x and y that satisfy both equations are x = 32.3 and y = 21,085.

Discussion

The matrix representation of a system of linear equations is a powerful tool for solving systems of linear equations. It allows us to represent the coefficients of the variables and the constants in a compact and organized way, making it easier to solve the system of equations. In this article, we used the matrix representation of a system of linear equations to solve a system of two linear equations with two variables. The values of x and y that satisfy both equations are x = 32.3 and y = 21,085.

Example

Here is an example of how to use the matrix representation of a system of linear equations to solve a system of three linear equations with three variables:

{ \begin{array}{l} 2x + 3y - z = 7 \\ x - 2y + 3z = -3 \\ 3x + 2y - z = 5 \end{array} \}

To solve this system of equations, we can use the following steps:

  1. Write the matrix equation: Write the matrix equation that represents the system of linear equations. In this case, the matrix equation is:

{ \begin{array}{l} 2x + 3y - z = 7 \\ x - 2y + 3z = -3 \\ 3x + 2y - z = 5 \end{array} \}

  1. Set the three equations equal to each other: Set the three equations equal to each other, since they all equal 0. This gives us:

2x+3y−z=0x−2y+3z=03x+2y−z=0{ 2x + 3y - z = 0 \\ x - 2y + 3z = 0 \\ 3x + 2y - z = 0 }

  1. Solve for x, y, and z: Solve for x, y, and z by isolating them on one side of the equation. We can do this by using the following steps:
  • Solve for x in the first equation:

2x+3y−z=0{ 2x + 3y - z = 0 }

2x=−3y+z{ 2x = -3y + z }

x=−32y+12z{ x = -\frac{3}{2}y + \frac{1}{2}z }

  • Solve for y in the second equation:

x−2y+3z=0{ x - 2y + 3z = 0 }

−32y+12z+2y−3z=0{ -\frac{3}{2}y + \frac{1}{2}z + 2y - 3z = 0 }

−12y−52z=0{ -\frac{1}{2}y - \frac{5}{2}z = 0 }

y=−5z{ y = -5z }

  • Solve for z in the third equation:

3x+2y−z=0{ 3x + 2y - z = 0 }

3(−32y+12z)+2y−z=0{ 3(-\frac{3}{2}y + \frac{1}{2}z) + 2y - z = 0 }

−92y+32z+2y−z=0{ -\frac{9}{2}y + \frac{3}{2}z + 2y - z = 0 }

−52y+12z=0{ -\frac{5}{2}y + \frac{1}{2}z = 0 }

z=5y{ z = 5y }

  1. Substitute the values of y and z into the equation for x: Substitute the values of y and z into the equation for x to get:

x=−32(−5z)+12(5z){ x = -\frac{3}{2}(-5z) + \frac{1}{2}(5z) }

x=152z+52z{ x = \frac{15}{2}z + \frac{5}{2}z }

x=202z{ x = \frac{20}{2}z }

x=10z{ x = 10z }

  1. Find the values of x, y, and z: Now that we have the values of x, y, and z, we can find the values of x, y, and z by substituting z into the equations for x and y. We will use the equation for x:

x=10z{ x = 10z }

Substituting z = 1 into this equation gives us:

x=10(1){ x = 10(1) }

x=10{ x = 10 }

We can also find the values of y and z by substituting z into the equations for y and z. We will use the equation for y:

y=−5z{ y = -5z }

Substituting z = 1 into this equation gives us:

y=−5(1){ y = -5(1) }

y=−5{ y = -5 }

We can also find the value of z by substituting x and y into the equation for z. We will use the equation for z:

z=5y{ z = 5y }

Substituting y = -5 into this equation gives us:

z=5(−5){ z = 5(-5) }

z=−25{ z = -25 }

Conclusion

In this article, we used a matrix to solve a system of three linear equations with three variables. We first wrote the matrix equation that represents the system of linear equations, then set the three equations equal to each other. We solved for x, y, and z by isolating them on one side of the equation, and then found the values of x, y, and z by substituting z into the equations for x and y. The values of x, y, and z that satisfy all three equations are x = 10, y = -5, and z = -25.

Discussion

The matrix representation of a system of linear equations is a powerful tool for solving systems of linear equations. It allows us to represent the coefficients of the variables and the constants in a compact and organized way, making it easier to solve the system of equations. In this article, we used the matrix representation of a system of linear equations to solve a system of three linear equations
Frequently Asked Questions about Solving Systems of Linear Equations using a Matrix

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Introduction

In the previous article, we discussed how to solve a system of linear equations using a matrix. In this article, we will answer some frequently asked questions about solving systems of linear equations using a matrix.

Q: What is a matrix?

A: A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In the context of solving a system of linear equations, a matrix can be used to represent the coefficients of the variables and the constants in the equations.

Q: How do I write a matrix equation?

A: To write a matrix equation, you need to represent the coefficients of the variables and the constants in a rectangular array. For example, if you have two equations with two variables, x and y, you can write the matrix equation as:

{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \}

This matrix equation represents the system of linear equations.

Q: How do I solve a system of linear equations using a matrix?

A: To solve a system of linear equations using a matrix, you need to follow these steps:

  1. Write the matrix equation: Write the matrix equation that represents the system of linear equations.
  2. Set the two equations equal to each other: Set the two equations equal to each other, since they both equal y.
  3. Solve for x: Solve for x by isolating it on one side of the equation.
  4. Find the value of y: Now that you have the value of x, you can find the value of y by substituting x into one of the original equations.

Q: What if I have a system of three linear equations with three variables?

A: If you have a system of three linear equations with three variables, you can use the same steps as before to solve the system of equations. However, you will need to use a 3x3 matrix to represent the coefficients of the variables and the constants in the equations.

Q: How do I find the values of x, y, and z in a system of three linear equations with three variables?

A: To find the values of x, y, and z in a system of three linear equations with three variables, you need to follow these steps:

  1. Write the matrix equation: Write the matrix equation that represents the system of linear equations.
  2. Set the three equations equal to each other: Set the three equations equal to each other, since they all equal 0.
  3. Solve for x, y, and z: Solve for x, y, and z by isolating them on one side of the equation.
  4. Find the values of x, y, and z: Now that you have the values of x, y, and z, you can find the values of x, y, and z by substituting z into the equations for x and y.

Q: What if I have a system of linear equations with more than three variables?

A: If you have a system of linear equations with more than three variables, you can use the same steps as before to solve the system of equations. However, you will need to use a larger matrix to represent the coefficients of the variables and the constants in the equations.

Q: How do I know if a system of linear equations has a solution?

A: A system of linear equations has a solution if the matrix equation has a unique solution. If the matrix equation has no solution or an infinite number of solutions, then the system of linear equations does not have a solution.

Q: What if I have a system of linear equations with no solution?

A: If you have a system of linear equations with no solution, then the matrix equation has no solution. This means that the system of linear equations is inconsistent, and there is no value of x, y, and z that satisfies all the equations in the system.

Q: What if I have a system of linear equations with an infinite number of solutions?

A: If you have a system of linear equations with an infinite number of solutions, then the matrix equation has an infinite number of solutions. This means that there are an infinite number of values of x, y, and z that satisfy all the equations in the system.

Conclusion

In this article, we answered some frequently asked questions about solving systems of linear equations using a matrix. We discussed how to write a matrix equation, how to solve a system of linear equations using a matrix, and how to find the values of x, y, and z in a system of three linear equations with three variables. We also discussed how to determine if a system of linear equations has a solution, and what to do if the system of linear equations has no solution or an infinite number of solutions.