Math Problem Statement
Solution
To solve the system of equations given in the image by graphing, we need to rewrite the equations in slope-intercept form (i.e., ) for easier graphing. Here are the two equations from the image:
Step 1: Solve for in each equation.
For the first equation:
Subtract and from both sides:
For the second equation:
Subtract and from both sides:
Step 2: Graph the equations.
Now that both equations are in the form , where is the slope and is the y-intercept:
- The first equation has a slope of and a y-intercept of .
- The second equation has a slope of and a y-intercept of .
Step 3: Find the point of intersection.
You can graph these two lines and find where they intersect. To solve this algebraically:
Set the two expressions for equal to each other:
Solve for :
Substitute into one of the original equations to find . Using :
Final Answer:
The solution is .
Would you like more details or explanations on any of these steps?
Here are some related questions to deepen your understanding:
- How do you interpret the slope of a line graphically?
- What are the benefits of solving systems of equations graphically compared to other methods?
- How would you verify the solution of a system of equations using substitution?
- Can all systems of linear equations be solved by graphing? Why or why not?
- How would you interpret a system of equations that has no solution or infinitely many solutions?
Tip: In a system of equations, graphing gives you a visual understanding of how the two lines relate, such as whether they are parallel, intersecting, or coincident (overlapping).
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Systems of Equations
Graphing
Formulas
Slope-intercept form y = mx + b
Substitution method
Theorems
Systems of linear equations and intersections
Suitable Grade Level
Grades 9-11