Math Problem Statement

Solve the system of equations: For Problem 4a: \[\begin{cases} y = 25 + 30x \\ y = 15 + 32x \end{cases}\] For Problem 6a: \[\begin{cases} y = 4 - 3x \\ 5x + 2y = -7 \end{cases}\]

Solution

Let's solve each of these systems of equations one at a time.

Problem 4a:

y = 25 + 30x \\ y = 15 + 32x \end{cases}$$ To solve this system, we can set the two equations for $$y$$ equal to each other because both expressions equal $$y$$: $$25 + 30x = 15 + 32x$$ Now, isolate $$x$$ by moving terms involving $$x$$ to one side and constants to the other: $$25 - 15 = 32x - 30x$$ $$10 = 2x$$ $$x = 5$$ Now substitute $$x = 5$$ back into one of the original equations to find $$y$$. We'll use $$y = 25 + 30x$$: $$y = 25 + 30(5)$$ $$y = 25 + 150$$ $$y = 175$$ So, the solution for Problem 4a is: $$(x, y) = (5, 175)$$ --- ### Problem 6a: $$\begin{cases} y = 4 - 3x \\ 5x + 2y = -7 \end{cases}$$ We can substitute the expression for $$y$$ from the first equation into the second equation. 1. Substitute $$y = 4 - 3x$$ into $$5x + 2y = -7$$: $$5x + 2(4 - 3x) = -7$$ 2. Distribute the $$2$$: $$5x + 8 - 6x = -7$$ 3. Combine like terms: $$-x + 8 = -7$$ 4. Subtract 8 from both sides: $$-x = -15$$ 5. Divide by -1: $$x = 15$$ Now, substitute $$x = 15$$ back into the first equation to find $$y$$: $$y = 4 - 3(15)$$ $$y = 4 - 45$$ $$y = -41$$ So, the solution for Problem 6a is: $$(x, y) = (15, -41)$$ --- Would you like more details on any part of the solution? ### Follow-Up Questions: 1. How would you solve these systems if both equations were in standard form? 2. Can you verify the solution by substituting both values back into the original equations? 3. What are other methods for solving systems of equations besides substitution? 4. How can you recognize when a system of equations has no solution or infinitely many solutions? 5. What are some applications of solving systems of equations in real-world scenarios? #### Tip: When solving a system by substitution, always double-check by plugging the solution back into both original equations to confirm accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Substitution Method
Linear Equations

Formulas

y = mx + b (linear equation form)
Setting equations equal to each other for substitution: if y = f(x) and y = g(x), then f(x) = g(x)

Theorems

Substitution Method for Solving Systems of Equations

Suitable Grade Level

Grades 8-10

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