System of Equations Study Guide

Overview

A system of equations is a set of two or more equations involving the same set of variables. The solutions to the system are the values of the variables that satisfy all the equations simultaneously.

Types of Systems

There are three types of systems of equations:

Consistent and Independent: The system has a unique solution.
Consistent and Dependent: The system has infinitely many solutions.
Inconsistent: The system has no solution.

Methods for Solving Systems of Equations

There are several methods for solving systems of equations, including:

Graphing: Plot the equations on the coordinate plane and find the point of intersection.
Substitution: Solve one equation for one variable and substitute the expression into the other equation.
Elimination: Add or subtract the equations to eliminate one of the variables.

Example

Consider the following system of equations:
2x + 3y = 11
4x - 2y = 6

To solve this system using the substitution method, we can solve the first equation for x:
2x = 11 - 3y
x = (11 - 3y)/2

Then, we substitute the expression for x into the second equation:
4((11 - 3y)/2) - 2y = 6
22 - 6y - 2y = 6
-8y = -16
y = 2

Finally, we can substitute the value of y back into the expression for x to find the complete solution.

Practice Problems

1. Solve the following system of equations using the graphing method:
2x + y = 5
3x - y = 7

2. Solve the following system of equations using the elimination method:
3x + 2y = 10
2x - 3y = -5

◂Math Worksheets and Study Guides Eighth Grade. Perimeter and area

The resources above cover the following skills:

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.

Use visualization, spatial reasoning, and geometric modeling to solve problems.

Use geometric models to represent and explain numerical and algebraic relationships.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.