Math Problem Statement

Graph the system below and write its solution: \( \begin{cases} y = \frac{1}{2}x - 2 \\ -2x + y = 4 \end{cases} \)

Solution

The system of equations given in the image is:

y = \frac{1}{2}x - 2 \\ -2x + y = 4 \end{cases}$$ ### Step 1: Graph the first equation The first equation is already in slope-intercept form $$y = mx + b$$, where: - $$m = \frac{1}{2}$$ is the slope, - $$b = -2$$ is the y-intercept. You can plot the y-intercept at $$(0, -2)$$ and use the slope $$\frac{1}{2}$$ to rise 1 unit and run 2 units to the right, marking another point at $$(2, -1)$$. Draw a line through these points. ### Step 2: Rewrite and graph the second equation The second equation is $$-2x + y = 4$$. Solve for $$y$$ to put it in slope-intercept form: $$y = 2x + 4$$ Here, the slope $$m = 2$$ and the y-intercept $$b = 4$$. Plot the y-intercept at $$(0, 4)$$ and use the slope $$2$$ to rise 2 units and run 1 unit to the right, marking another point at $$(1, 6)$$. Draw a line through these points. ### Step 3: Find the solution To find the solution to this system, we need to find where the two lines intersect. By solving algebraically, set the two expressions for $$y$$ equal to each other: $$\frac{1}{2}x - 2 = 2x + 4$$ Multiply both sides by 2 to eliminate the fraction: $$x - 4 = 4x + 8$$ Now solve for $$x$$: $$x - 4x = 8 + 4$$ $$-3x = 12$$ $$x = -4$$ Substitute $$x = -4$$ back into one of the original equations, say $$y = \frac{1}{2}x - 2$$: $$y = \frac{1}{2}(-4) - 2 = -2 - 2 = -4$$ Thus, the solution is $$(-4, -4)$$. This is the point where the two lines intersect, and it is the solution to the system. ### Conclusion: The solution to the system is $$(-4, -4)$$. Would you like more details on any part of this explanation? Here are 5 related questions you could explore: 1. How do you convert an equation to slope-intercept form? 2. What does the slope of a line represent in real-world contexts? 3. How can you check the solution to a system of equations? 4. What are different methods to solve systems of equations (e.g., substitution, elimination)? 5. How does the graph of a line change when you alter its slope? **Tip:** When graphing lines, always plot at least two points to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Graphing
Linear Equations
Systems of Equations

Formulas

Slope-intercept form y = mx + b
Solving systems of linear equations

Theorems

Intersection of lines as the solution to a system of equations

Suitable Grade Level

Grades 8-10