Math Problem Statement

Solve the following system by the method of substitution: 3x + 5y = 5 2x + y = 1

Solution

The system of equations provided is:

3x + 5y &= 5 \\ 2x + y &= 1 \end{aligned}$$ We are asked to solve it using the method of substitution. Here's the step-by-step process: ### Step 1: Solve one equation for one variable. We'll solve the second equation $$2x + y = 1$$ for $$y$$: $$y = 1 - 2x$$ ### Step 2: Substitute the expression for $$y$$ into the first equation. Now, substitute $$y = 1 - 2x$$ into the first equation $$3x + 5y = 5$$: $$3x + 5(1 - 2x) = 5$$ ### Step 3: Simplify and solve for $$x$$. Distribute the $$5$$ on the left-hand side: $$3x + 5 - 10x = 5$$ Combine like terms: $$-7x + 5 = 5$$ Subtract 5 from both sides: $$-7x = 0$$ Solve for $$x$$: $$x = 0$$ ### Step 4: Substitute $$x = 0$$ back into the expression for $$y$$. Now substitute $$x = 0$$ into $$y = 1 - 2x$$: $$y = 1 - 2(0) = 1$$ ### Final Solution: The solution to the system is $$x = 0$$ and $$y = 1$$. ### Answer: $$(x, y) = (0, 1)$$ Would you like more details, or do you have any questions? Here are 5 related questions to consider: 1. What if the system of equations had no solutions? 2. How do we solve a system using the elimination method instead of substitution? 3. Can we solve a system graphically, and how would that work? 4. What happens if the equations are dependent (infinite solutions)? 5. How does solving a system with three variables differ from two variables? Tip: Always check your solution by substituting the values of $$x$$ and $$y$$ back into both original equations!

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Substitution Method
Systems of Equations

Formulas

Solving for y in terms of x: y = 1 - 2x
Substituting in another equation

Theorems

The method of substitution for solving systems of linear equations

Suitable Grade Level

Grades 9-10

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