Graphing Linear Systems: Solving No-Solution Systems with Parallel Lines

Math Problem Statement

Graph the system and write its solution: -6x - 2y = 6, y = -3x - 2.

The system of equations provided in the image is:

-6x - 2y = 6 \\ y = -3x - 2 \end{cases}$$ ### Step 1: Simplify the first equation The first equation can be simplified by dividing the entire equation by $$-2$$ to put it in slope-intercept form: $$-6x - 2y = 6 \quad \text{(Divide by -2)} \quad 3x + y = -3$$ Thus, the first equation becomes: $$y = -3x - 3$$ ### Step 2: Compare both equations The system now consists of: 1. $$y = -3x - 3$$ 2. $$y = -3x - 2$$ These are two parallel lines since both have the same slope $$-3$$, but different y-intercepts ($$-3$$ and $$-2$$). ### Step 3: Conclusion Parallel lines never intersect, which means there is **no solution** to this system of equations. Therefore, the correct solution is **"No solution."** --- Would you like more details, or do you have any questions? Here are 5 related questions you might find interesting: 1. How do we determine if two lines are parallel by looking at their equations? 2. What is the significance of the slope and y-intercept in a linear equation? 3. How can we verify graphically that two lines do not intersect? 4. How does solving a system of equations change when using substitution versus graphing? 5. What does it mean when a system of equations has "infinitely many solutions"? **Tip:** When two equations in a system have the same slope but different y-intercepts, the system will always have no solution because the lines are parallel.

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Algebra
Linear Equations
Graphing Systems of Equations

y = mx + b
Solving systems of linear equations

Parallel Line Theorem

Grades 8-10

Solve a System of Equations by Graphing: y = -3x - 3 and 6x + 2y = -2

Graphing Systems of Linear Equations: y = x - 6

Solve a System of Linear Equations: Graphing Parallel Lines

No Solution in the System of Equations y = 3x + 2 and y = 3x - 6

Graphing System of Linear Equations: y = 3x - 3 and 6x - 2y = 6