Math Problem Statement

Solve the simultaneous equations 2a - 3b = 7 and 3a + 2b = 4.

Solution

The given system of simultaneous equations is:

2a - 3b = 7 \tag{1} 3a + 2b = 4 \tag{2}

We can solve this using either substitution or elimination. I'll use elimination here for simplicity.


Step 1: Eliminate one variable

To eliminate bb, we'll multiply both equations by suitable coefficients to make the coefficients of bb equal in magnitude. Multiply (1) by 2 and (2) by 3:

4a - 6b = 14 \tag{3} 9a + 6b = 12 \tag{4}

Now add the two equations:

(4a6b)+(9a+6b)=14+12(4a - 6b) + (9a + 6b) = 14 + 12 13a=2613a = 26 a=2a = 2


Step 2: Solve for bb

Substitute a=2a = 2 into either of the original equations. Using equation (1):

2(2)3b=72(2) - 3b = 7 43b=74 - 3b = 7 3b=3-3b = 3 b=1b = -1


Final Answer:

The solution is: a=2,b=1a = 2, \, b = -1


Would you like further clarification or additional methods? Here are related questions you might consider:

  1. What are the advantages of solving by elimination compared to substitution?
  2. How would the solution change if the second equation were 3a+2b=63a + 2b = 6?
  3. Can we verify the solution by substituting back into both equations?
  4. How can these techniques extend to three-variable systems?
  5. What are the practical applications of solving such equations?

Tip: Always verify your solution by substituting back into the original equations!

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Math Problem Analysis

Mathematical Concepts

Simultaneous Equations
Linear Equations
Elimination Method

Formulas

Linear equation format: ax + by = c

Theorems

Basic properties of equality

Suitable Grade Level

Grades 8-10