Solve a System of Equations using Substitution Method: -x + y = 1 and 4x

Math Problem Statement

Use the substitution method to solve the system of equations: -x + y = 1 and 4x - 3y = -2.

Let's solve the given system of equations using the substitution method. The system is:

-x + y = 1 \\ 4x - 3y = -2 \end{cases}$$ ### Step 1: Solve one equation for one variable From the first equation: $$-x + y = 1 \implies y = x + 1$$ (we solved for $$y$$). ### Step 2: Substitute into the second equation Now substitute $$y = x + 1$$ into the second equation: $$4x - 3(x + 1) = -2$$ ### Step 3: Solve for $$x$$ Distribute the $$-3$$ on the left-hand side: $$4x - 3x - 3 = -2$$ Simplify: $$x - 3 = -2$$ Add 3 to both sides: $$x = 1$$ ### Step 4: Solve for $$y$$ Now that we know $$x = 1$$, substitute it back into $$y = x + 1$$: $$y = 1 + 1 = 2$$ ### Final Solution: The solution to the system is $$x = 1$$ and $$y = 2$$. ### Conclusion: Since we found a single unique solution, the answer is **"One solution"** with the solution pair $$(x, y) = (1, 2)$$. Do you have any questions or need further details on this method? ### Related Questions: 1. How can you check if a system has no solution or infinite solutions using the substitution method? 2. How does the substitution method differ from the elimination method? 3. What happens if you try solving a system where both equations represent the same line? 4. Can substitution be used in systems with three variables? 5. How do you recognize dependent and independent systems of equations? ### Tip: Always substitute the solution back into both original equations to ensure it's correct.

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Algebra
Linear Equations
Systems of Equations
Substitution Method

Solving for y: -x + y = 1 ⟹ y = x + 1
Substitution into second equation: 4x - 3(x + 1) = -2

Unique Solution in Systems of Equations

Grades 9-10

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