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Consistent and Dependent Systems

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Consistent and Dependent Systems

Study Guide

Key Definition

A system of linear equations is consistent if it has at least one solution (exactly one or infinitely many), and is dependent if it has infinitely many solutions. For example, consider the system $y = 2x + 5$ and $y = 4x + 3$. These two lines intersect at exactly one point, illustrating a consistent and independent system.

Important Notes

If a system has no solution, it is said to be inconsistent.
In an inconsistent system, the graphs of the lines are parallel and do not intersect, so there is no solution.
In a consistent and dependent system, both equations represent the same line.

Mathematical Notation

$y = mx + b$ is the equation of a line, where m is the slope and b is the y-intercept.

Why It Works

The system of equations works by finding the values that satisfy both equations at the same time. These values form an ordered pair which is the solution of the system.

Remember

The solution to a system of equations is the ordered pair that satisfies both equations.

Quick Reference

Consistent system:$\text{At least one solution}$

Dependent system:$\text{Infinite number of solutions}$

Inconsistent system:$\text{No solution}$

Understanding Consistent and Dependent Systems

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A consistent system of equations is one that has at least one solution. This means that the lines representing the equations intersect at exactly one point or infinitely many points.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Consider the system of equations $y = 2x + 3$ and $y = 2x + 3$. How would you classify this system?

Real-World Problem

Question Exercise

Intermediate

Skateboard Sales

You and your friend start selling skateboards. You both sell the same number of skateboards, but your friend charges $5 more per skateboard. After a week, your friend has made $100 more than you. Can you find out how many skateboards you each sold?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the system of equations $y = 2x + 3$ and $y = 2x + 5$. How would you classify this system and why?

Challenge Quiz

Single Choice Quiz

Advanced

Consider the system of equations $y = 3x + 2$ and $2y = 6x + 4$. How would you classify this system?

Recap

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