Math Problem Statement
Solve the system of linear equations: y = -1x - 1 and y = 2x - 7 using a graphing utility.
Solution
The problem asks you to solve the following system of linear equations by graphing:
y &= -1x - 1 \\ y &= 2x - 7 \end{aligned}$$ We can solve this system by graphing the two equations and finding the intersection point, as that represents the solution where both equations are satisfied. ### Steps to graph the equations: 1. **First equation: $$y = -x - 1$$** - Slope: $$-1$$ (which means for every step right, go down 1 unit) - Y-intercept: $$-1$$ (the point where the line crosses the y-axis) - Plot the y-intercept (0, -1) and use the slope to find additional points. 2. **Second equation: $$y = 2x - 7$$** - Slope: $$2$$ (for every step right, go up 2 units) - Y-intercept: $$-7$$ - Plot the y-intercept (0, -7) and use the slope to find additional points. ### Finding the intersection: When graphing both lines, they intersect at a single point. This point is the solution to the system of equations. Alternatively, solving algebraically by setting the two equations equal to each other: $$-x - 1 = 2x - 7$$ Solving for $$x$$: $$-x - 1 = 2x - 7$$ $$-1 - (-7) = 2x + x \quad \Rightarrow \quad 6 = 3x$$ $$x = 2$$ Substitute $$x = 2$$ into one of the original equations to find $$y$$: $$y = 2(2) - 7 = 4 - 7 = -3$$ Thus, the solution is: $$(x, y) = (2, -3)$$ Would you like further details or have any questions? --- Here are 5 related questions you could explore: 1. How can you verify the solution to a system of equations? 2. What are other methods to solve systems of equations besides graphing? 3. What is the geometric interpretation of the solution to a system of equations? 4. How would you solve the system if both lines were parallel? 5. How can a system of equations have no solution or infinitely many solutions? **Tip:** Always double-check your graph by plugging the solution back into the original equations to ensure it satisfies both.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Systems of Equations
Formulas
y = mx + b
-1x - 1 = 2x - 7
Theorems
Intersection of Lines Theorem
Suitable Grade Level
Grades 8-10