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:

  1. ax+by=18ax + by = 18
  2. axby=6ax - by = 6

The point of intersection is given as (3,2)(3, -2). To find aa and bb, we can substitute x=3x = 3 and y=2y = -2 into both equations.

Step 1: Substituting into the first equation

a(3)+b(2)=18a(3) + b(-2) = 18 This simplifies to: 3a2b=18(Equation 1)3a - 2b = 18 \quad \text{(Equation 1)}

Step 2: Substituting into the second equation

a(3)b(2)=6a(3) - b(-2) = 6 This simplifies to: 3a+2b=6(Equation 2)3a + 2b = 6 \quad \text{(Equation 2)}

Now, we have a system of equations:

  1. 3a2b=183a - 2b = 18
  2. 3a+2b=63a + 2b = 6

Step 3: Solving the system of equations

Add the two equations to eliminate bb: (3a2b)+(3a+2b)=18+6(3a - 2b) + (3a + 2b) = 18 + 6 This simplifies to: 6a=246a = 24 a=4a = 4

Step 4: Substituting a=4a = 4 into one of the original equations

We can substitute a=4a = 4 into either Equation 1 or 2. Let's use Equation 1: 3(4)2b=183(4) - 2b = 18 122b=1812 - 2b = 18 2b=1812-2b = 18 - 12 2b=6-2b = 6 b=3b = -3

Final Answer:

The values of aa and bb are: a=4,b=3a = 4, \quad b = -3

Would you like more details on this solution or have any questions?


Here are 5 related questions to expand your understanding:

  1. How do you solve a system of linear equations with more than two variables?
  2. How would the solution change if the point of intersection was different?
  3. How can we interpret the slopes of these two lines geometrically?
  4. How do we graph the lines ax+by=18ax + by = 18 and axby=6ax - by = 6 after finding aa and bb?
  5. 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