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:
SYSTEM B:
- 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:
Now substitute from the first equation into this equation.
From :
Substituting into :
Thus, the solution for SYSTEM A is .
Step 2: Check the Options
Now, let's rewrite the second equation of SYSTEM B in terms of its equivalent form:
Option A:
Option B:
Option C:
Option D:
We need to convert each option to check if it provides the same solutions as SYSTEM A.
Step 3: Check Each Option
-
Option A:
-
Option B:
-
Option C:
-
Option D: 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: .
Would you like further details on any step or additional questions? Here are some related questions you might find interesting:
- How do you solve a system of equations using substitution?
- What are the conditions for two systems of equations to be equivalent?
- How can you graphically represent these systems?
- What is the significance of the coefficients in determining equivalence?
- 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
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