Math Problem Statement

3x+2y=45 4x -y=5

Solution

You've provided two linear equations:

  1. 3x+2y=453x + 2y = 45
  2. 4xy=54x - y = 5

This is a system of linear equations in two variables, xx and yy. We can solve this system using various methods, such as substitution, elimination, or matrix methods (like the determinant method). Let's use the substitution method here.

Step 1: Solve the second equation for yy

From the second equation: 4xy=54x - y = 5 y=4x5y = 4x - 5

Step 2: Substitute yy in the first equation

Now, substitute y=4x5y = 4x - 5 into the first equation: 3x+2(4x5)=453x + 2(4x - 5) = 45 Simplify and solve for xx: 3x+8x10=453x + 8x - 10 = 45 11x10=4511x - 10 = 45 11x=5511x = 55 x=5x = 5

Step 3: Solve for yy

Now that we know x=5x = 5, substitute it back into the expression for yy: y=4(5)5y = 4(5) - 5 y=205=15y = 20 - 5 = 15

Solution

The solution to the system is: x=5,y=15x = 5, \quad y = 15

This means the point (5,15)(5, 15) is the intersection of the two lines represented by the equations.

Would you like to explore another method to solve this system or have any other questions?


Here are 5 related questions you might find interesting:

  1. What are the advantages of using the substitution method over the elimination method?
  2. How can you solve the system of equations using the elimination method?
  3. What does it mean if the system of equations has no solution or infinitely many solutions?
  4. How can you solve a system of linear equations using matrices?
  5. What is the graphical interpretation of the solution to a system of linear equations?

Tip: Always check your solution by substituting the values of xx and yy back into the original equations to verify they satisfy both equations.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Substitution Method

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12