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.

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.

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.

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.

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

Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

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

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

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

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.