Solving Systems of Linear Equations (2024)

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9 Questions

What is a system of linear equations?

A set of two or more linear equations that must be true at the same time.

Which of the following methods is used to solve systems of linear equations?

All of the above

What is the slope-intercept form of a linear equation?

y = mx + b

What is the purpose of the slope-intercept form of a linear equation?

To graph linear equations and identify the slope and y-intercept

What is the standard form of a linear equation?

Ax + By = C

What is the purpose of the standard form of a linear equation?

To write linear equations in a consistent format and solve systems of linear equations

How can the standard form of a linear equation be converted to slope-intercept form?

By rearranging the equation to isolate y and dividing by the coefficient of y (if necessary)

What is the first step in the substitution method for solving systems of linear equations?

Solve one equation for one variable

What is the final step in the elimination method for solving systems of linear equations?

Substitute the value of the variable back into one of the original equations to find the value of the other variable

Study Notes

Solving Systems Of Equations

A system of linear equations is a set of two or more linear equations that must be true at the same time.
Systems of equations can be solved using:
- Substitution method
  - Elimination method
- Graphical method

Substitution Method

Solve one equation for one variable
Substitute the expression into the other equation
Solve for the other variable
Substitute the value of the variable back into one of the original equations to find the value of the other variable

Elimination Method

Make the coefficients of one variable opposites
Add the equations to eliminate the variable
Solve for the remaining variable
Substitute the value of the variable back into one of the original equations to find the value of the other variable

Slope-Intercept Form

The slope-intercept form of a linear equation is given by:
- y = mx + b
- m is the slope (rate of change)
- b is the y-intercept (point where the line crosses the y-axis)
Slope-intercept form is useful for:
- Graphing linear equations
- Identifying the slope and y-intercept of a line

Standard Form

The standard form of a linear equation is given by:
- Ax + By = C
- A, B, and C are integers
- A and B are not both zero
Standard form is useful for:
- Writing linear equations in a consistent format
- Solving systems of linear equations
Standard form can be converted to slope-intercept form by:
- Rearranging the equation to isolate y
- Dividing by the coefficient of y (if necessary)

Systems of Linear Equations

A system consists of two or more linear equations that must be true at the same time
Can be solved using three methods: Substitution, Elimination, and Graphical method

Solving Systems of Equations using Substitution Method

Solve one equation for one variable
Substitute the expression into the other equation to solve for the other variable
Substitute the value of the variable back into one of the original equations to find the value of the other variable

Solving Systems of Equations using Elimination Method

Make the coefficients of one variable opposites
Add the equations to eliminate the variable
Solve for the remaining variable
Substitute the value of the variable back into one of the original equations to find the value of the other variable

Slope-Intercept Form

Equation is given by: y = mx + b
m represents the slope (rate of change)
b represents the y-intercept (point where the line crosses the y-axis)
Useful for graphing linear equations and identifying the slope and y-intercept of a line

Standard Form

Equation is given by: Ax + By = C
A, B, and C are integers
A and B are not both zero
Useful for writing linear equations in a consistent format and solving systems of linear equations
Can be converted to slope-intercept form by rearranging the equation to isolate y and dividing by the coefficient of y (if necessary)

Learn how to solve systems of linear equations using substitution, elimination, and graphical methods. Discover the steps to find the solution to a system of equations.

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Solving Systems of Linear Equations (2024)

FAQs

Can you always check your answers when solving a linear equation? ›

The meaning of the solution value is that it is the x-value that makes the equation true. So, to check your answer, you plug your solution value back into the original equation, and you make sure that the equation "works" with that value.

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What are the possible answers for a system of linear equations? ›

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line). This article reviews all three cases. One solution. A system of linear equations has one solution when the graphs intersect at a point.

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What's the easiest way to solve systems of linear equations? ›

SOLVE A SYSTEM OF EQUATIONS BY SUBSTITUTION.

Solve one of the equations for either variable.
Substitute the expression from Step 1 into the other equation.
Solve the resulting equation.
Substitute the solution in Step 3 into either of the original equations to find the other variable.

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Nov 24, 2022

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How to find out how many solutions a system of equations has? ›

A system of two equations can be classified as follows: If the slopes are the same but the y-intercepts are different, the system has no solution. If the slopes are different, the system has one solution. If the slopes are the same and the y-intercepts are the same, the system has infinitely many solutions.

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What is the trick to solving linear equations? ›

To solve linear equations, find the value of the variable that makes the equation true. Use the inverse of the number that multiplies the variable, and multiply or divide both sides by it. Simplify the result to get the variable value. Check your answer by plugging it back into the equation.

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How to check your answer when solving a system of equations? ›

To check a system of equations by substitution, you plug your values for x and y into the original equations. If both simplified expressions are true then your answer is correct.

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How to tell if a system of equations has no solution or infinitely many? ›

If we can solve the equation and get something like x=b where b is a specific number, then we have one solution. If we end up with a statement that's always false, like 3=5, then there's no solution. If we end up with a statement that's always true, like 5=5, then there are infinite solutions..

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What is the infinite number of solutions? ›

An infinite solution can be produced if the lines are coincident and they must have the same y-intercept. The two lines having the same y-intercept and the slope, are the exact same line. In simpler words, we can say that if the two lines are sharing the same line, then the system would result in an infinite solution.

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What is the fastest way to solve linear equations? ›

Elimination method is obviously the best way to solve 2 simultaneous linear equations. Eliminate (cancel) 1 variable from both equations. Find the value of the other variable.

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What is the best method to solve the linear system? ›

The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows: In the first equation, solve for one of the variables in terms of the others. Substitute this expression into the remaining equations.

What are the 3 main types of ways to solve a system of linear equations? ›

There are three ways to solve systems of linear equations in two variables:

graphing.
substitution method.
elimination method.

Jul 10, 2011

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Is it possible for a system of linear equations to have no solutions? ›

A system of two linear equations has no solution if the lines are parallel. Parallel lines on a coordinate plane have the same slope and different y-intercepts (see figure 3 for an example of this).

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Does 0 0 mean infinite solutions? ›

If you get an equation that is always true, such as 0 = 0, then there are infinite solutions.

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How to tell if a system of equations has one solution without graphing? ›

If the slope of the lines is different, then the system of equations has one solution. Therefore, you can determine that a system of linear equations has one solution without graphing by comparing the slope of the lines.

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What are the rules for solving linear equations? ›

The following steps provide a good method to use when solving linear equations.

Simplify each side of the equation by removing parentheses and combining like terms.
Use addition or subtraction to isolate the variable term on one side of the equation.
Use multiplication or division to solve for the variable.

Do you have to check your answers for absolute value equations? ›

It is essential to check the answers you find to figure out if they are extraneous. To check if any of your roots are extraneous, plug each of the roots back in to the original equation.

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