Solving a System of Linear Equations with Substitution

Math Problem Statement

Solve the system of equations: y = -0.5x + 1 and 5x + y = -2.

The system of equations in the image is:

y = -0.5x + 1 \\ 5x + y = -2 \end{cases}$$ ### Step 1: Substitute $$y = -0.5x + 1$$ into the second equation. Substituting $$y$$ in the second equation: $$5x + (-0.5x + 1) = -2$$ Simplify this equation: $$5x - 0.5x + 1 = -2$$ $$4.5x + 1 = -2$$ ### Step 2: Solve for $$x$$. Subtract 1 from both sides: $$4.5x = -3$$ Divide both sides by 4.5: $$x = \frac{-3}{4.5} = -\frac{2}{3}$$ ### Step 3: Solve for $$y$$. Now, substitute $$x = -\frac{2}{3}$$ into the first equation $$y = -0.5x + 1$$: $$y = -0.5\left(-\frac{2}{3}\right) + 1 = \frac{1}{3} + 1 = \frac{4}{3}$$ ### Final Answer: The solution to the system is $$\left(-\frac{2}{3}, \frac{4}{3}\right)$$, but this does not match any of the given options. Upon closer inspection, this suggests that none of the provided answers correctly match the calculated solution. The system has a single solution, but it's not one of the options. Do you want any further details or explanations on the steps? ### Related Questions: 1. How can I verify the solution of a system of equations? 2. What does it mean if a system of equations has no solutions? 3. How do I identify if a system has infinitely many solutions? 4. Can you explain how substitution works in solving systems of equations? 5. What are other methods to solve a system of linear equations besides substitution? ### Tip: Always double-check the given options after solving an equation, as sometimes the correct solution might not match exactly with the given answers.

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Algebra
Linear Systems
Substitution Method

Substitution method for solving systems of linear equations

Unique solution theorem for linear systems
Consistency of linear systems

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