Math Problem Statement

Solve the system by graphing: 2x + 4y = 8 and 3x - 4y = 12.

Solution

The image contains a system of linear equations:

2x + 4y &= 8 \\ 3x - 4y &= 12 \end{aligned}$$ You are asked to solve the system by graphing and select the solution. ### Steps to solve the system by graphing: 1. **Rewrite both equations in slope-intercept form** (y = mx + b). For the first equation: $$2x + 4y = 8 \implies 4y = -2x + 8 \implies y = -\frac{1}{2}x + 2$$ The slope is $$-\frac{1}{2}$$ and the y-intercept is 2. For the second equation: $$3x - 4y = 12 \implies -4y = -3x + 12 \implies y = \frac{3}{4}x - 3$$ The slope is $$\frac{3}{4}$$ and the y-intercept is -3. 2. **Plot both equations** on the graph using their slopes and intercepts: - For $$y = -\frac{1}{2}x + 2$$, start at (0, 2) and use the slope $$-\frac{1}{2}$$, meaning go down 1 unit and right 2 units. - For $$y = \frac{3}{4}x - 3$$, start at (0, -3) and use the slope $$\frac{3}{4}$$, meaning go up 3 units and right 4 units. 3. **Find the intersection point**, which represents the solution of the system. After plotting both lines, you should find that the two lines intersect at a specific point, which will be the solution to the system of equations. Would you like a detailed graph or any further clarification? Here are five related questions you might explore: 1. How do you solve systems of equations algebraically (e.g., by substitution or elimination)? 2. How do slopes and intercepts affect the shape and position of a line in a graph? 3. Can two lines in a system of equations be parallel? If so, what does that mean for the solution? 4. How do you determine if a system of equations has infinitely many solutions? 5. What methods can be used to solve nonlinear systems of equations? **Tip:** When graphing a system of linear equations, always double-check that you have plotted enough points accurately for each line to ensure their intersection (if any) is clear.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Graphing Systems of Equations
Slope-Intercept Form

Formulas

Slope-intercept form y = mx + b
Rewriting equations in slope-intercept form: 2x + 4y = 8 -> y = -1/2x + 2 and 3x - 4y = 12 -> y = 3/4x - 3

Theorems

Intersection of two lines gives the solution to a system of linear equations

Suitable Grade Level

Grade 8-10

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