8 Questions
What is the first step in solving a system of linear equations using the substitution method?
Solve one equation for one variable
What is the purpose of performing row operations when using the matrix method to solve a system of linear equations?
To transform the matrix into upper triangular form
In the elimination method, why do we multiply the equations by necessary multiples?
To make the coefficients of one variable the same
What is the last step in solving a system of linear equations using the substitution method?
Back-substitute to find the value of the first variable
How do we write a system of linear equations as an augmented matrix for the matrix method?
[A | b]
What do we do after eliminating one variable using the elimination method?
Solve for the remaining variable, then back-substitute to find the value of the eliminated variable
Why do we need to solve for one variable first when using the substitution method?
To substitute the expression into the other equation(s)
What is the advantage of using the matrix method to solve a system of linear equations?
It can be easily extended to solve systems with more than two variables
Study Notes
System of Linear Equations
A system of linear equations consists of two or more linear equations with variables and coefficients.
Substitution Method
- Solve one equation for one variable
- Substitute the expression into the other equation(s)
- Solve for the remaining variable(s)
- Back-substitute to find the value of the first variable
Example: 2x + 3y = 7 x - 2y = -3
- Solve the first equation for x: x = 7 - 3y
- Substitute into the second equation: (7 - 3y) - 2y = -3
- Solve for y, then back-substitute to find x
Matrix Method
- Write the system as an augmented matrix: [A | b]
- Perform row operations to transform the matrix into upper triangular form
- Solve for the variables by back-substitution
Example: 2x + 3y = 7 x - 2y = -3
- Augmented matrix: [[2, 3 | 7], [1, -2 | -3]]
- Perform row operations to get the matrix into upper triangular form
- Solve for x and y by back-substitution
Elimination Method
- Multiply the equations by necessary multiples such that the coefficients of one variable are the same
- Add or subtract the equations to eliminate one variable
- Solve for the remaining variable
- Back-substitute to find the value of the eliminated variable
Example: 2x + 3y = 7 x - 2y = -3
- Multiply the first equation by 1 and the second equation by 2: 2x + 3y = 7, 2x - 4y = -6
- Subtract the equations to eliminate x: 7y = 13
- Solve for y, then back-substitute to find x
System of Linear Equations
- A system of linear equations consists of two or more linear equations with variables and coefficients.
Substitution Method
- To solve a system of linear equations using the substitution method, solve one equation for one variable.
- Substitute the expression into the other equation(s).
- Solve for the remaining variable(s).
- Back-substitute to find the value of the first variable.
Matrix Method
- Write the system as an augmented matrix: [A | b].
- Perform row operations to transform the matrix into upper triangular form.
- Solve for the variables by back-substitution.
Elimination Method
- Multiply the equations by necessary multiples such that the coefficients of one variable are the same.
- Add or subtract the equations to eliminate one variable.
- Solve for the remaining variable.
- Back-substitute to find the value of the eliminated variable.
- The elimination method involves making the coefficients of one variable the same, then adding or subtracting the equations to eliminate that variable.
Learn how to solve systems of linear equations using the substitution method, a step-by-step approach to finding variable values.
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