Math Problem Statement
solve each system of linear equations by graphing.
Solve each system of linear equations by graphing.
Solution
Let's work on solving the systems of linear equations through graphing. I'll address each problem from the image you provided one by one:
11.
System of equations:
-2x + y &= 6 \\ y &= -x \end{aligned}$$ To solve this by graphing: 1. **First equation**: Rearrange into slope-intercept form $$y = mx + b$$: $$y = 2x + 6$$ The slope is $$2$$, and the y-intercept is $$6$$. 2. **Second equation**: Already in slope-intercept form $$y = -x$$, where the slope is $$-1$$ and the y-intercept is $$0$$. ### 12. System of equations: $$\begin{aligned} x - 3y &= -6 \\ y &= \frac{2}{3}x - 1 \end{aligned}$$ 1. **First equation**: Solve for $$y$$: $$3y = x + 6 \quad \text{or} \quad y = \frac{1}{3}x + 2$$ This gives us a slope of $$\frac{1}{3}$$ and a y-intercept of $$2$$. 2. **Second equation**: Already in slope-intercept form $$y = \frac{2}{3}x - 1$$, where the slope is $$\frac{2}{3}$$ and the y-intercept is $$-1$$. ### 13. System of equations: $$\begin{aligned} y &= 2x - 3 \\ x + 3y &= 12 \end{aligned}$$ 1. **First equation**: Already in slope-intercept form $$y = 2x - 3$$, with a slope of $$2$$ and a y-intercept of $$-3$$. 2. **Second equation**: Solve for $$y$$: $$3y = -x + 12 \quad \text{or} \quad y = -\frac{1}{3}x + 4$$ The slope is $$-\frac{1}{3}$$, and the y-intercept is $$4$$. ### 14. System of equations: $$\begin{aligned} 5x + 4y &= -24 \\ y &= \frac{3}{4}x + 2 \end{aligned}$$ 1. **First equation**: Solve for $$y$$: $$4y = -5x - 24 \quad \text{or} \quad y = -\frac{5}{4}x - 6$$ The slope is $$-\frac{5}{4}$$, and the y-intercept is $$-6$$. 2. **Second equation**: Already in slope-intercept form $$y = \frac{3}{4}x + 2$$, with a slope of $$\frac{3}{4}$$ and a y-intercept of $$2$$. #### To Graph Each System: 1. **Plot the lines for each equation** on the grid provided. 2. **Look for the intersection point**. This point represents the solution to the system of equations, where both equations are satisfied. Would you like me to graph one or more of these systems for you? --- ### Expansion Questions: 1. What is the meaning of the slope and y-intercept in the context of a linear equation? 2. How can you determine whether two lines are parallel or perpendicular? 3. What happens if the two lines in a system of equations are parallel? 4. What does it mean geometrically if a system of equations has no solution? 5. How can you check your graphing solution algebraically? ### Tip: When graphing lines, ensure you accurately plot the y-intercept first and then use the slope to find other points along the line.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Linear Equations
Graphing
Formulas
Slope-intercept form: y = mx + b
Standard form: Ax + By = C
Theorems
Two lines intersect at a point if they have different slopes.
Parallel lines have the same slope but no intersection.
Perpendicular lines have slopes that are negative reciprocals of each other.
Suitable Grade Level
Grades 8-10