Math Problem Statement
the graphs of ax+by=18 and ax-by=6 intersect at (3,-2). Find a and b
Solution
We are given two linear equations:
The point of intersection is given as . To find and , we can substitute and into both equations.
Step 1: Substituting into the first equation
This simplifies to:
Step 2: Substituting into the second equation
This simplifies to:
Now, we have a system of equations:
Step 3: Solving the system of equations
Add the two equations to eliminate : This simplifies to:
Step 4: Substituting into one of the original equations
We can substitute into either Equation 1 or 2. Let's use Equation 1:
Final Answer:
The values of and are:
Would you like more details on this solution or have any questions?
Here are 5 related questions to expand your understanding:
- How do you solve a system of linear equations with more than two variables?
- How would the solution change if the point of intersection was different?
- How can we interpret the slopes of these two lines geometrically?
- How do we graph the lines and after finding and ?
- What is the method of elimination, and how does it apply to solving systems of equations?
Tip: Always check your solutions by substituting back into the original equations to verify they satisfy both.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
System of Linear Equations
Formulas
ax + by = c
Elimination method
Theorems
Linear equation theorem
Suitable Grade Level
Grades 9-12