Math Problem Statement

11x–2y=5 4x+6y=9 What value of c will make this system have the same solution? 11x–2y=5 cx=24

Solution

To find the value of cc that will make the system:

  1. 11x2y=511x - 2y = 5
  2. cx=24cx = 24

have the same solution as the original system:

  1. 11x2y=511x - 2y = 5
  2. 4x+6y=94x + 6y = 9,

we need to express the second equation in terms of only xx, so we eliminate yy in both systems.

Step 1: Solve the second equation of the original system for yy

From the original system, the second equation is: 4x+6y=94x + 6y = 9

Solve for yy: 6y=94x6y = 9 - 4x y=94x6y = \frac{9 - 4x}{6} y=322x3y = \frac{3}{2} - \frac{2x}{3}

Step 2: Substitute this expression for yy in the first equation of the original system

Now substitute y=322x3y = \frac{3}{2} - \frac{2x}{3} into the first equation 11x2y=511x - 2y = 5:

11x2(322x3)=511x - 2\left( \frac{3}{2} - \frac{2x}{3} \right) = 5 Simplify: 11x2×32+2×2x3=511x - 2 \times \frac{3}{2} + 2 \times \frac{2x}{3} = 5 11x3+4x3=511x - 3 + \frac{4x}{3} = 5 Multiply everything by 3 to eliminate the fraction: 33x9+4x=1533x - 9 + 4x = 15 37x=2437x = 24 x=2437x = \frac{24}{37}

Step 3: Substitute this value of xx into the equation cx=24cx = 24

Now, from the second system, we know that cx=24cx = 24. Substituting x=2437x = \frac{24}{37} into this equation:

c×2437=24c \times \frac{24}{37} = 24 Multiply both sides by 37: c×24=24×37c \times 24 = 24 \times 37 c=37c = 37

Thus, the value of cc that will make the system have the same solution is 37\boxed{37}.


Would you like more details on any step?

Here are 5 related questions for further exploration:

  1. How can we verify that x=2437x = \frac{24}{37} satisfies both systems?
  2. How would the solution change if the second equation had a different constant, such as cx=12cx = 12?
  3. How can you solve this system graphically?
  4. What happens if the value of cc is different from 37? Does the system still have a solution?
  5. Can you solve this system using substitution or elimination?

Tip: When solving systems of equations with different forms, always try to eliminate one variable first for clarity.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Substitution Method

Formulas

Linear Equation: ax + by = c
Solving a system using substitution

Theorems

Consistency of a system of linear equations
Substitution Theorem

Suitable Grade Level

Grades 9-12