Math Problem Statement

For each ordered pair, determine whether it is a solution to the system of equations: 2x + 5y = -9 and y = 4x + 7.

Solution

To determine whether each ordered pair is a solution to the given system of equations:

1. & \quad 2x + 5y = -9 \\ 2. & \quad y = 4x + 7 \end{aligned}$$ We substitute the $$x$$ and $$y$$ values from each pair into both equations to check if they satisfy both simultaneously. --- ### Steps: 1. **(4, 6):** - Substituting into $$2x + 5y = -9$$: $$2(4) + 5(6) = 8 + 30 = 38 \neq -9$$ This does not satisfy the first equation. - **Not a solution.** 2. **(3, -3):** - Substituting into $$2x + 5y = -9$$: $$2(3) + 5(-3) = 6 - 15 = -9$$ This satisfies the first equation. - Substituting into $$y = 4x + 7$$: $$-3 = 4(3) + 7 \implies -3 = 12 + 7 \implies -3 \neq 19$$ This does not satisfy the second equation. - **Not a solution.** 3. **(-2, -1):** - Substituting into $$2x + 5y = -9$$: $$2(-2) + 5(-1) = -4 - 5 = -9$$ This satisfies the first equation. - Substituting into $$y = 4x + 7$$: $$-1 = 4(-2) + 7 \implies -1 = -8 + 7 \implies -1 = -1$$ This satisfies the second equation. - **This is a solution.** 4. **(0, 7):** - Substituting into $$2x + 5y = -9$$: $$2(0) + 5(7) = 0 + 35 = 35 \neq -9$$ This does not satisfy the first equation. - **Not a solution.** --- ### Final Table: | $$(x, y)$$ | Is it a solution? | |------------|-------------------| | (4, 6) | No | | (3, -3) | No | | (-2, -1) | Yes | | (0, 7) | No | Would you like further details or clarification? 😊 --- Here are 5 related questions to expand your knowledge: 1. How do we graphically represent systems of equations? 2. What are the methods to solve systems of equations (e.g., substitution, elimination)? 3. What does it mean if a system of equations has no solution? 4. How can we identify whether a system of equations is dependent or independent? 5. Can systems of equations be solved using matrices? **Tip:** Always check both equations when solving systems to verify consistency!

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Substitution Method

Formulas

2x + 5y = -9
y = 4x + 7

Theorems

Consistency of Systems of Equations

Suitable Grade Level

Grades 8-10

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