Math Problem Statement

Use the substitution method to solve the system of equations: -x + y = 2 4x + 3y = -7

Solution

To solve the system of equations using the substitution method, we start with the system given in the image:

-x + y &= 2 \quad \text{(1)} \\ 4x + 3y &= -7 \quad \text{(2)} \end{aligned}$$ ### Step 1: Solve for one variable from one equation From equation (1), we can easily solve for $$y$$: $$y = x + 2$$ ### Step 2: Substitute the expression for $$y$$ into the second equation Now, substitute $$y = x + 2$$ into equation (2): $$4x + 3(x + 2) = -7$$ Simplify the equation: $$4x + 3x + 6 = -7$$ $$7x + 6 = -7$$ ### Step 3: Solve for $$x$$ Now, solve for $$x$$: $$7x = -7 - 6$$ $$7x = -13$$ $$x = \frac{-13}{7}$$ ### Step 4: Substitute $$x = -\frac{13}{7}$$ back into the expression for $$y$$ Now, substitute $$x = -\frac{13}{7}$$ back into $$y = x + 2$$: $$y = -\frac{13}{7} + 2 = -\frac{13}{7} + \frac{14}{7} = \frac{1}{7}$$ ### Final Answer: The solution is $$x = -\frac{13}{7}$$ and $$y = \frac{1}{7}$$. Since there is a unique solution, the answer to the question is: - One solution: $$(-\frac{13}{7}, \frac{1}{7})$$ Would you like any further details or explanations? --- Here are 5 follow-up questions to deepen your understanding: 1. How does the substitution method differ from the elimination method in solving systems of equations? 2. What would happen if both equations led to a contradiction, such as $$0 = 5$$? What would the solution set look like? 3. Can the system of equations be solved graphically, and if so, what would the graphical solution represent? 4. What would change in this system if the coefficients of $$x$$ and $$y$$ were altered slightly? Would the solution change drastically? 5. How can systems of equations with more than two variables be solved using substitution? **Tip:** Always check your solutions by substituting them back into both original equations to verify that they satisfy both.

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Math Problem Analysis

Mathematical Concepts

Linear Systems of Equations
Substitution Method

Formulas

y = x + 2 (from -x + y = 2)
Substitution into 4x + 3(x + 2) = -7

Theorems

Unique Solution for Linear Systems

Suitable Grade Level

Grades 9-12