How do you solve systems of linear equations using substitution, elimination, and graphing methods?

Solving systems of linear equations is an important topic in Algebra. There are three main methods to solve systems of linear equations: substitution, elimination, and graphing.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting the resulting expression into the other equation. The goal is to eliminate one of the variables and solve for the other. Here are the steps:

• Solve one of the equations for one of the variables.
• Substitute the expression from step 1 into the other equation.
• Solve the resulting equation for the remaining variable.
• Substitute the value from step 3 into either equation to find the value of the other variable.
• Check your answer by substituting both values into both equations.

Elimination Method

The elimination method involves adding or subtracting the equations to eliminate one of the variables. Here are the steps:

• Multiply one or both equations by a constant so that the coefficients of one of the variables are additive inverses.
• Add or subtract the equations to eliminate one of the variables.
• Solve the resulting equation for the remaining variable.
• Substitute the value from step 3 into either equation to find the value of the other variable.
• Check your answer by substituting both values into both equations.

Graphing Method

The graphing method involves graphing the two equations on the same coordinate plane and identifying the point of intersection. Here are the steps:

• Graph both equations on the same coordinate plane.
• Identify the point of intersection.
• Write the solution as an ordered pair (x, y) where x and y are the coordinates of the point of intersection.

Each of these methods has its own advantages and disadvantages. The substitution method is useful when one of the equations is already solved for one of the variables. The elimination method is useful when neither equation is solved for a variable or when the coefficients of one variable are already additive inverses. The graphing method is useful when the equations are simple and can be graphed easily.

There are three main methods for solving systems of linear equations: substitution, elimination, and graphing.

Substitution

The substitution method involves solving one of the equations for one of the variables, and then substituting that value into the other equation. This will result in a single equation with one unknown, which can be solved using the usual methods.

For example, consider the system of equations:

x + y = 5
2x - y = 3

We can solve the first equation for y:

y = 5 - x

Substituting this into the second equation, we get:

2x - (5 - x) = 3

Simplifying, we get:

x = 4

Substituting this back into the first equation, we get:

4 + y = 5

Solving for y, we get:

y = 1

Therefore, the solution to the system of equations is (4, 1).

Elimination

The elimination method involves adding or subtracting the equations in such a way that one of the variables is eliminated. This will result in a single equation with one unknown, which can be solved using the usual methods.

For example, consider the system of equations:

x + y = 5
2x - y = 3

We can eliminate y by adding the equations together, since the y-terms will cancel out:

3x = 8

Solving for x, we get:

x = 8/3

Substituting this back into either of the original equations, we can find that y = 1. Therefore, the solution to the system of equations is (8/3, 1).

Graphing

The graphing method involves graphing each of the equations on a coordinate plane. The solution to the system of equations is the point where the two lines intersect.

For example, consider the system of equations:

x + y = 5
2x - y = 3

The first equation can be graphed as a line with a slope of 1 and a y-intercept of 5. The second equation can be graphed as a line with a slope of -2 and a y-intercept of 3. The two lines intersect at the point (4, 1). Therefore, the solution to the system of equations is (4, 1).

Each of these methods can be used to solve systems of linear equations. The best method to use will depend on the specific system of equations.