Math Problem Statement

4x + 3y = 11 2x - 2y = -12 Solve each system by elimination.

Solution

We are given the system of equations:

4x + 3y = 11 \\ 2x - 2y = -12 \end{cases}$$ To solve this system by elimination, we'll follow these steps: ### Step 1: Multiply the equations to align coefficients We need to eliminate one variable. Let’s eliminate $$x$$ by making the coefficients of $$x$$ in both equations equal. The first equation has $$4x$$, and the second equation has $$2x$$. We can multiply the second equation by 2 to make the coefficients of $$x$$ equal in both equations: Multiply the second equation by 2: $$2(2x - 2y = -12) \quad \Rightarrow \quad 4x - 4y = -24$$ Now the system is: $$\begin{cases} 4x + 3y = 11 \\ 4x - 4y = -24 \end{cases}$$ ### Step 2: Subtract the equations to eliminate $$x$$ Now subtract the second equation from the first to eliminate $$x$$: $$(4x + 3y) - (4x - 4y) = 11 - (-24)$$ Simplify: $$4x - 4x + 3y + 4y = 11 + 24$$ $$7y = 35$$ ### Step 3: Solve for $$y$$ Now, solve for $$y$$: $$y = \frac{35}{7} = 5$$ ### Step 4: Substitute $$y = 5$$ into one of the original equations Substitute $$y = 5$$ into the first equation $$4x + 3y = 11$$: $$4x + 3(5) = 11$$ Simplify: $$4x + 15 = 11$$ Subtract 15 from both sides: $$4x = 11 - 15 = -4$$ Solve for $$x$$: $$x = \frac{-4}{4} = -1$$ ### Final Answer: The solution to the system is: $$x = -1, \quad y = 5$$ Would you like to check this solution or explore more examples? Here are 5 related questions: 1. What would happen if both equations already had the same $$y$$-coefficients? 2. Can you solve the same system using substitution instead of elimination? 3. How would you solve this system if the second equation was $$3x - 2y = 5$$? 4. How can you graph the two equations and find the solution visually? 5. Can you solve the system if one equation is in a different form, like $$y = mx + b$$? **Tip:** When using elimination, always check if you can simplify both equations first to make the elimination step easier.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Elimination Method

Formulas

Linear equation form ax + by = c

Theorems

Elimination method

Suitable Grade Level

Grades 8-10

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