Math Problem Statement

Please view the different systems to answer the question: SYSTEM A: 5x + 2y = -3 3x + 3y = 9
SYSTEM B: 3x + 3y = 9 ? Which of the following terms will make the systems equivalent? A. -8x – 5y = 6 B. 8x + 5y = 6 C. -10x + 4y = 6 D. -10x – 4y = -6

Solution

To determine which of the options will make SYSTEM B equivalent to SYSTEM A, we need to express both systems in a form that allows for direct comparison.

SYSTEM A:

  1. 5x+2y=35x + 2y = -3
  2. 3x+3y=93x + 3y = 9

SYSTEM B:

  1. 3x+3y=93x + 3y = 9
  2. We need to find a second equation from the given options.

Step 1: Solve SYSTEM A

From the second equation of SYSTEM A, we can simplify it:

3x+3y=9    y=x+33x + 3y = 9 \implies y = -x + 3

Now substitute yy from the first equation into this equation.

From 5x+2y=35x + 2y = -3:

5x+2(x+3)=3    5x2x+6=3    3x+6=3    3x=9    x=35x + 2(-x + 3) = -3 \implies 5x - 2x + 6 = -3 \implies 3x + 6 = -3 \implies 3x = -9 \implies x = -3

Substituting x=3x = -3 into y=x+3y = -x + 3:

y=(3)+3=3+3=6y = -(-3) + 3 = 3 + 3 = 6

Thus, the solution for SYSTEM A is (x,y)=(3,6)(x, y) = (-3, 6).

Step 2: Check the Options

Now, let's rewrite the second equation of SYSTEM B in terms of its equivalent form:

Option A: 8x5y=6-8x - 5y = 6

Option B: 8x+5y=68x + 5y = 6

Option C: 10x+4y=6-10x + 4y = 6

Option D: 10x4y=6-10x - 4y = -6

We need to convert each option to check if it provides the same solutions as SYSTEM A.

Step 3: Check Each Option

  1. Option A: 8x5y=6    8x+5y=6 (not equivalent)-8x - 5y = 6 \implies 8x + 5y = -6 \text{ (not equivalent)}

  2. Option B: 8x+5y=6    5y=8x+6    y=85x65 (not equivalent)8x + 5y = 6 \implies -5y = -8x + 6 \implies y = \frac{8}{5}x - \frac{6}{5} \text{ (not equivalent)}

  3. Option C: 10x+4y=6    10x4y=6 (not equivalent)-10x + 4y = 6 \implies 10x - 4y = -6 \text{ (not equivalent)}

  4. Option D: 10x4y=6    10x+4y=6-10x - 4y = -6 \implies 10x + 4y = 6 This simplifies similarly and has the same structure as one of SYSTEM A's equations when multiplied by 2.

Conclusion

The option that will make SYSTEM B equivalent to SYSTEM A is D: 10x4y=6-10x - 4y = -6.

Would you like further details on any step or additional questions? Here are some related questions you might find interesting:

  1. How do you solve a system of equations using substitution?
  2. What are the conditions for two systems of equations to be equivalent?
  3. How can you graphically represent these systems?
  4. What is the significance of the coefficients in determining equivalence?
  5. How do you convert a standard equation to slope-intercept form?

Tip: Always check if you can manipulate the equations into the same format for easier comparison!

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Math Problem Analysis

Mathematical Concepts

Systems of Equations
Linear Algebra

Formulas

Equivalence of Linear Systems
Substitution Method

Theorems

Condition for Equivalent Systems

Suitable Grade Level

Grades 9-12