Math Problem Statement

True or False: (5, 2) is a solution to the following system? y = (2/5)x + 8, y = (3/5)x - 5

Solution

The question asks whether the point (5,2)(5, 2) is a solution to the following system of equations:

y=25x+8y = \frac{2}{5}x + 8 y=35x5y = \frac{3}{5}x - 5

To determine if (5,2)(5, 2) is a solution, we can substitute x=5x = 5 and y=2y = 2 into both equations and check if they satisfy both.

First equation:

y=25x+8y = \frac{2}{5}x + 8 Substituting x=5x = 5 and y=2y = 2: 2=25(5)+82 = \frac{2}{5}(5) + 8 2=2+8(which simplifies to)2=102 = 2 + 8 \quad \text{(which simplifies to)} \quad 2 = 10 This is false, so the point (5,2)(5, 2) does not satisfy the first equation.

Since it fails the first equation, the point (5,2)(5, 2) is not a solution to the system.

The answer is False.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

  1. How do you solve a system of equations graphically?
  2. What is the significance of a solution in a system of linear equations?
  3. Can a system of equations have no solution? When does that occur?
  4. How do you check if a point satisfies a quadratic system of equations?
  5. What methods are available to solve systems of linear equations (e.g., substitution, elimination)?

Tip: When checking if a point is a solution to a system of equations, always substitute the point into both equations and ensure both are satisfied.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

y = (2/5)x + 8
y = (3/5)x - 5

Theorems

Substitution method
Solution to a system of linear equations

Suitable Grade Level

Grades 8-10