What is the solution to the system of equations x + y = 10 and x - y = 2?

The solution is the ordered pair (6, 4), which satisfies both equations.

Explain what it means for a system of equations to be inconsistent.

An inconsistent system has no solution, which occurs when the lines representing the equations are parallel and never intersect.

If one equation in a system is already solved for a variable, which method is most efficient to use?

The substitution method is most efficient when one equation is already solved for a variable.

Riddle: I have two equations but no solution; my lines never meet. What am I?

An inconsistent system of equations.

Fill in the blank: The method that combines two equations to eliminate one variable is called ___ method.

Elimination.

What indicates that a system of equations has infinitely many solutions?

When the equations represent the same line, leading to an identity like 0 = 0.

Using elimination, solve the system: 2x + 3y = 8 and 3x - y = 1. What is the solution?

The solution is (1, 2).

True or False: A consistent and dependent system has exactly one solution. If False, explain.

False. A consistent and dependent system has infinitely many solutions.

What does the graphical representation of a system of equations show when the lines intersect at exactly one point?

It shows that the system is consistent and independent, indicating one unique solution.

What is the graphical representation of a system of equations with one solution?

Two lines intersect uniquely.

• Indicates a unique solution.

• Represents two equations.

• Graphically visualizes intersection.

Fill in the blank: A system of equations is called __________ when there are no solutions.

Inconsistent

Explain the substitution method for solving systems of equations.

Substitution method simplifies equations.

• Solve one equation first.

• Substitute into the other.

• Solve for the remaining variable.

True or False: When two lines representing a system of equations are identical, the system has a unique solution.

False. When two lines are identical, the system has infinitely many solutions.

What are the steps to solve a system of equations using the elimination method?

Steps to solve using elimination

• Write equations in standard form.

• Adjust equations for opposing coefficients.

• Eliminate one variable, solve and substitute.

Riddle: I am a method that uses a graph to find the point where two lines meet, what am I?

Graphing method

Fill in the blank: The solution to a system of equations is an ordered pair that __________ both equations simultaneously.

Satisfies

What special case occurs when a system of equations results in a contradiction like 0 = 5?

This indicates that the system has no solution, meaning the lines are parallel.

Describe the steps for solving a system of equations by graphing.

Steps for Solving Equations

• Rewrite equations in slope-intercept form.

• Graph each equation on the same axes.

• Identify intersection point for solution.