Math Problem Statement

3x + 7y = 16 4x - 3y = 9

Solution

We are given the system of linear equations:

3x+7y=16(Equation 1)3x + 7y = 16 \quad \text{(Equation 1)} 4x3y=9(Equation 2)4x - 3y = 9 \quad \text{(Equation 2)}

We can solve this system by using the substitution method or elimination method. Let's use the elimination method to eliminate one variable.

Step 1: Multiply the equations to align the coefficients

We want to eliminate one of the variables by making the coefficients of either xx or yy equal in both equations. Let's eliminate yy.

To do that, we'll multiply Equation 1 by 3 and Equation 2 by 7 to make the coefficients of yy opposites of each other:

3(3x+7y)=3(16)9x+21y=48(Equation 3)3(3x + 7y) = 3(16) \quad \Rightarrow \quad 9x + 21y = 48 \quad \text{(Equation 3)} 7(4x3y)=7(9)28x21y=63(Equation 4)7(4x - 3y) = 7(9) \quad \Rightarrow \quad 28x - 21y = 63 \quad \text{(Equation 4)}

Step 2: Add the two equations to eliminate yy

Now, add Equation 3 and Equation 4:

(9x+21y)+(28x21y)=48+63(9x + 21y) + (28x - 21y) = 48 + 63 9x+28x=1119x + 28x = 111 37x=11137x = 111

Step 3: Solve for xx

Now, solve for xx:

x=11137=3x = \frac{111}{37} = 3

Step 4: Substitute x=3x = 3 back into one of the original equations

Now that we know x=3x = 3, substitute this value back into either Equation 1 or Equation 2. Let's substitute it into Equation 1:

3(3)+7y=163(3) + 7y = 16 9+7y=169 + 7y = 16 7y=1697y = 16 - 9 7y=77y = 7 y=77=1y = \frac{7}{7} = 1

Final Solution:

Thus, the solution to the system of equations is: x=3andy=1x = 3 \quad \text{and} \quad y = 1

Would you like further details on how this solution was found? Let me know if you have any questions!


Here are some related questions to expand your understanding:

  1. What is the substitution method for solving a system of linear equations?
  2. How can you use matrix operations to solve a system of equations?
  3. What happens if the system of equations has no solution or infinitely many solutions?
  4. How can you graphically represent the solution of a system of linear equations?
  5. What is the concept of "linear independence" in a system of equations?

Tip: When using the elimination method, ensure that the coefficients of the variable you wish to eliminate are opposites. This can often simplify the process.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Elimination Method
Substitution Method

Formulas

Elimination method
Substitution method

Theorems

Linear systems solutions

Suitable Grade Level

Grades 8-10