Math Problem Statement

Solve the system by graphing: { 2x + y = 6, y = 4x - 3 }.

Solution

Let's solve the system of equations by graphing:

2x + y = 6 \\ y = 4x - 3 \end{cases}$$ ### Step 1: Convert to Slope-Intercept Form The second equation is already in slope-intercept form: $$y = 4x - 3$$ where the slope is **4** and the y-intercept is **-3**. For the first equation: $$2x + y = 6$$ Subtract $$2x$$ from both sides: $$y = -2x + 6$$ Now, this is in slope-intercept form, where the slope is **-2** and the y-intercept is **6**. ### Step 2: Graph Both Equations 1. **Graph $$y = 4x - 3$$** - Start at $$(0, -3)$$ on the y-axis. - Use the slope $$4$$ (rise 4, run 1) to plot another point at $$(1,1)$$. - Continue plotting points following the same slope. 2. **Graph $$y = -2x + 6$$** - Start at $$(0,6)$$ on the y-axis. - Use the slope $$-2$$ (down 2, right 1) to plot another point at $$(1,4)$$. - Continue plotting points following the same slope. ### Step 3: Find the Intersection The two lines intersect at the point **(3, 9)**. Thus, the solution to the system is **$$x = 3$$, $$y = 9$$**. Would you like a more detailed breakdown or a graph illustration? ### Related Questions: 1. How do you determine if a system of equations has no solution or infinite solutions? 2. What happens if both equations in a system have the same slope? 3. How do you solve a system of equations using substitution? 4. How do you solve a system of equations using elimination? 5. What real-world problems can be solved using systems of linear equations? **Tip:** When graphing, using different colors for each equation helps visualize the intersection more clearly.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Graphing Systems of Equations

Formulas

Slope-intercept form: y = mx + b

Theorems

Slope and Intercept Theorem

Suitable Grade Level

Grades 8-10

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