Solve the System of Linear Equations 2x + y = 10, -x

Math Problem Statement

Solve the system of linear equation {2x+y=10,-x-2y=15

To solve the system of linear equations:

1) \ \ 2x + y &= 10 \\ 2) \ \ -x - 2y &= 15 \end{aligned}$$ ### Step 1: Solve one equation for one variable Let's solve equation $$(1)$$ for $$y$$: $$2x + y = 10 \implies y = 10 - 2x$$ ### Step 2: Substitute into the other equation Substitute $$y = 10 - 2x$$ into equation $$(2)$$: $$-x - 2(10 - 2x) = 15$$ Now simplify: $$-x - 20 + 4x = 15$$ $$3x - 20 = 15$$ ### Step 3: Solve for $$x$$ Add 20 to both sides: $$3x = 35$$ Now, divide by 3: $$x = \frac{35}{3}$$ ### Step 4: Substitute $$x$$ back into the equation for $$y$$ Substitute $$x = \frac{35}{3}$$ into $$y = 10 - 2x$$: $$y = 10 - 2\left(\frac{35}{3}\right)$$ $$y = 10 - \frac{70}{3}$$ $$y = \frac{30}{3} - \frac{70}{3} = \frac{-40}{3}$$ ### Final Solution The solution to the system is: $$x = \frac{35}{3}, \quad y = \frac{-40}{3}$$ Would you like more details or have any questions? --- Here are 5 related questions: 1. How can substitution be used for more complex systems of equations? 2. What is the elimination method, and how could it be applied here? 3. How would you interpret these results geometrically? 4. How do you handle systems of equations with no solutions? 5. What is the matrix method for solving systems of linear equations? **Tip:** Always double-check your substitution steps to avoid small arithmetic errors that can lead to incorrect results.

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Algebra
Systems of Linear Equations
Substitution Method

Solving for a variable in terms of another: y = 10 - 2x
Substitution into another equation: -x - 2(10 - 2x) = 15

Linear Equation Theorem
Substitution Method

Grades 9-11

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