Math Problem Statement

Graph the system below and write its solution: 2x + y = -5 y = -(1/4)x + 2

Solution

You provided a system of equations that needs to be graphed and solved. The system is:

2x + y = -5 \\ y = -\frac{1}{4}x + 2 \end{cases}$$ ### Steps to Solve: 1. **Rewrite the first equation** to express $$y$$ in terms of $$x$$: $$y = -2x - 5$$ So, the system becomes: $$\begin{cases} y = -2x - 5 \\ y = -\frac{1}{4}x + 2 \end{cases}$$ 2. **Graph both equations**: - The first equation $$y = -2x - 5$$ is a line with a slope of $$-2$$ and a y-intercept at $$(0, -5)$$. - The second equation $$y = -\frac{1}{4}x + 2$$ is a line with a slope of $$-\frac{1}{4}$$ and a y-intercept at $$(0, 2)$$. 3. **Find the intersection**: The solution to the system is where the two lines intersect. We can set the two equations equal to each other: $$-2x - 5 = -\frac{1}{4}x + 2$$ 4. **Solve algebraically**: Multiply through by 4 to eliminate the fraction: $$4(-2x - 5) = -x + 8$$ $$-8x - 20 = -x + 8$$ Add $$8x$$ to both sides: $$-20 = 7x + 8$$ Subtract 8 from both sides: $$-28 = 7x$$ Divide by 7: $$x = -4$$ 5. **Find $$y$$**: Substitute $$x = -4$$ into one of the original equations (let's use $$y = -2x - 5$$): $$y = -2(-4) - 5 = 8 - 5 = 3$$ The solution is $$(x, y) = (-4, 3)$$. Would you like a graph of this system as well? ### Follow-up questions: 1. How do we find the slope of a linear equation? 2. What is the general form of a linear equation in slope-intercept form? 3. What does the slope of a line represent in a graph? 4. How do you interpret the solution of a system of linear equations? 5. How would you check the solution by substituting the values back into the original equations? **Tip:** Always verify your solution by substituting the values back into both original equations to ensure they hold true.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Graphing Systems of Equations

Formulas

y = mx + b (Slope-Intercept Form)
2x + y = -5
y = - (1/4)x + 2

Theorems

Intersection of Lines
Linear System of Equations

Suitable Grade Level

Grades 8-10

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