Math Problem Statement
Solve each system of linear equations by graphing.
Solve each system of linear equations by graphing.
Solution
To solve the given system of linear equations by graphing, we'll follow the steps below. The system of equations in the image is:
1) &\ -2x + y = 6 \\ 2) &\ y = -x \end{aligned}$$ ### Step 1: Rearrange the equations into slope-intercept form $$(y = mx + b)$$ - For equation (1), $$-2x + y = 6$$, rearrange to solve for $$y$$: $$y = 2x + 6$$ So, the first equation is $$y = 2x + 6$$, where the slope $$m = 2$$ and the y-intercept $$b = 6$$. - For equation (2), $$y = -x$$, it is already in slope-intercept form, where the slope $$m = -1$$ and the y-intercept $$b = 0$$. ### Step 2: Plot the lines 1. **Graph $$y = 2x + 6$$:** - Start at the y-intercept (0, 6). - Use the slope $$m = 2$$, which means "rise 2, run 1". From (0, 6), go up 2 units and right 1 unit to the point (1, 8). Continue this pattern to plot more points if needed. 2. **Graph $$y = -x$$:** - Start at the y-intercept (0, 0). - The slope is $$m = -1$$, meaning "fall 1, run 1". From (0, 0), go down 1 unit and right 1 unit to the point (1, -1). Continue this pattern to plot more points. ### Step 3: Find the intersection point The solution to the system of equations is where the two lines intersect. By plotting both lines, we find that they intersect at the point $$(-2, 2)$$. ### Solution The solution to the system of equations is: $$x = -2, \quad y = 2$$ This is the point where both equations are satisfied. Would you like further details or clarification on any of the steps? Here are five follow-up questions to deepen your understanding: 1. How do you check the solution algebraically by substituting $$(-2, 2)$$ into both equations? 2. What is the significance of the slope in interpreting the graph of a linear equation? 3. How would you solve the system using substitution or elimination methods instead of graphing? 4. Can all systems of equations be solved by graphing? Why or why not? 5. What does it mean if two lines are parallel in terms of solutions to the system? **Tip:** Always ensure that both lines are plotted accurately when solving systems by graphing to find the exact point of intersection.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Graphing
Formulas
y = mx + b (Slope-Intercept Form)
Slope = (change in y) / (change in x)
Theorems
Intersection of two lines as the solution to a system of linear equations
Suitable Grade Level
Grades 8-10
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