Linear Systems: Elimination

The elimination method involves adding or subtracting equations to eliminate one variable, allowing you to solve for the remaining variable.

Steps for the elimination method:

1. Add or subtract equations to eliminate one variable
2. Solve for the remaining variable
3. Substitute the value into one of the original equations

Example: Solving 2x + 3y = 11 and -2x + 5y = 13

1. Add the equations to eliminate x: 8y = 24
2. Solve for y: y = 3
3. Substitute y = 3 into 2x + 3y = 11 to find x = 1 The solution is (1, 3)

Highlight: You can multiply one or both equations by a constant to facilitate elimination.

Vocabulary: Consistent independent systems have exactly one solution, which can be found using the elimination method.

Linear Systems: Substitution

The substitution method involves isolating a variable in one equation and substituting it into the other equation.

Steps for the substitution method:

1. Isolate a variable in one equation
2. Substitute the expression into the other equation and solve
3. Use the result to find the value of the other variable

Example: Solving y = 3x + 2 and x + 2y = 11

1. Substitute y = 3x + 2 into x + 2y = 11
2. Solve for x: x + 2$3x + 2$ = 11, resulting in x = 1
3. Find y by substituting x = 1 into y = 3x + 2, giving y = 5 The solution is (1, 5)

Highlight: The substitution method is particularly useful when one equation is already solved for a variable.

Vocabulary: A consistent independent system will have a unique solution that can be found using the substitution method.

Linear Systems: Graphing

Linear systems consist of two or more linear equations with the same variables. A solution to a linear system is an ordered pair that satisfies all equations in the system. When graphing, the intersection point of the lines represents the solution.

To solve a linear system using the graph and check method:

1. Graph both equations on the same coordinate plane
2. Estimate the coordinates of the intersection point
3. Check the coordinates algebraically

Definition: A linear system with exactly one solution is called a consistent independent system.

Example: Solving the system -x + y = -7 and x + 4y = -8

1. Graph both equations
2. Identify the intersection point $4, -3$
3. Check: -4 + $-3$ = -7 and 4 + 4$-3$ = -8 The solution is $4, -3$

Highlight: Always verify the solution by substituting it back into both original equations.