Math Problem Statement

{ 4x+5y=23 2x−3y=−1 ​

Solution

To solve the system of linear equations:

4x+5y=23(1)4x + 5y = 23 \quad (1) 2x3y=1(2)2x - 3y = -1 \quad (2)

we can use either the substitution method or the elimination method. Here, I'll use the elimination method.

Step 1: Align the equations:

4x+5y=23(1)4x + 5y = 23 \quad (1) 2x3y=1(2)2x - 3y = -1 \quad (2)

Step 2: Make the coefficients of xx or yy equal:

Multiply equation (2) by 2 so that the coefficient of xx in both equations becomes the same:

4x+5y=23(1)4x + 5y = 23 \quad (1) 4x6y=2(3)4x - 6y = -2 \quad (3)

Step 3: Subtract equation (3) from equation (1):

(4x+5y)(4x6y)=23(2)(4x + 5y) - (4x - 6y) = 23 - (-2) 4x+5y4x+6y=23+24x + 5y - 4x + 6y = 23 + 2 11y=2511y = 25 y=2511y = \frac{25}{11}

Step 4: Substitute yy back into one of the original equations to solve for xx:

Substitute y=2511y = \frac{25}{11} into equation (2):

2x3(2511)=12x - 3\left(\frac{25}{11}\right) = -1 2x7511=12x - \frac{75}{11} = -1 2x=1+75112x = -1 + \frac{75}{11} 2x=11+75112x = \frac{-11 + 75}{11} 2x=64112x = \frac{64}{11} x=3211x = \frac{32}{11}

So, the solution to the system of equations is:

x=3211,y=2511x = \frac{32}{11}, \quad y = \frac{25}{11}

Would you like more details or have any questions?


Here are some related questions for further practice:

  1. Solve the system of equations using the substitution method: 3x+4y=173x + 4y = 17 and x2y=3x - 2y = -3.
  2. Find the solution to the system: 5x+2y=85x + 2y = 8 and 3x4y=143x - 4y = -14.
  3. Determine xx and yy for the equations: x+y=10x + y = 10 and 2xy=32x - y = 3.
  4. Solve the system: 7x+3y=17x + 3y = 1 and xy=4x - y = 4.
  5. Use elimination to solve: 2x+5y=162x + 5y = 16 and 4x10y=84x - 10y = -8.
  6. Solve by substitution: x+2y=6x + 2y = 6 and 4xy=54x - y = 5.
  7. Determine the solution: 3xy=73x - y = 7 and x+2y=4x + 2y = 4.
  8. Solve using any method: 6x+y=196x + y = 19 and 2x3y=72x - 3y = -7.

Tip: When solving systems of equations, it's often useful to double-check your solutions by substituting them back into the original equations to ensure they satisfy both equations.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Elimination Method

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12