Math Problem Statement

Solve the system of equations: 3x + 2y = 16, 5x - 4y = 10. Solve for x and y using the method of your choice (substitution or elimination). Don’t forget to check your solution by substituting back into the original system.

Solution

We are solving the system of equations:

3x + 2y = 16 \\ 5x - 4y = 10 \end{cases}$$ We can solve this using the **elimination method** (you could also use substitution if preferred). Here's the step-by-step process: --- ### 1. Elimination of one variable We aim to eliminate $$y$$ first. To do this, we need to make the coefficients of $$y$$ in both equations match in absolute value. - The first equation is $$3x + 2y = 16$$ → Multiply through by $$2$$ to get: $$6x + 4y = 32$$ - The second equation is $$5x - 4y = 10$$. We now have the system: $$\begin{cases} 6x + 4y = 32 \\ 5x - 4y = 10 \end{cases}$$ --- ### 2. Add the equations Add the two equations together to eliminate $$y$$: $$(6x + 4y) + (5x - 4y) = 32 + 10$$ Simplify: $$6x + 5x + 4y - 4y = 42$$ $$11x = 42$$ Solve for $$x$$: $$x = \frac{42}{11}$$ --- ### 3. Solve for $$y$$ Substitute $$x = \frac{42}{11}$$ into one of the original equations. Let's use $$3x + 2y = 16$$: $$3\left(\frac{42}{11}\right) + 2y = 16$$ Simplify $$3 \cdot \frac{42}{11}$$: $$\frac{126}{11} + 2y = 16$$ Convert $$16$$ to a fraction with denominator $$11$$: $$\frac{126}{11} + 2y = \frac{176}{11}$$ Subtract $$\frac{126}{11}$$ from both sides: $$2y = \frac{176}{11} - \frac{126}{11}$$ Simplify: $$2y = \frac{50}{11}$$ Divide by $$2$$: $$y = \frac{50}{11} \cdot \frac{1}{2}$$ $$y = \frac{25}{11}$$ --- ### 4. Solution The solution is: $$x = \frac{42}{11}, \quad y = \frac{25}{11}.$$ --- ### 5. Verification Substitute $$x = \frac{42}{11}$$ and $$y = \frac{25}{11}$$ back into both original equations to confirm they are satisfied. --- Do you need help with verification, or would you like a detailed explanation of another method? Let me know! ### Related Questions: 1. How do you solve a system of equations using the substitution method? 2. What is the graphical interpretation of solving systems of equations? 3. How can elimination be used for systems with no solution or infinite solutions? 4. Solve another system: $$2x + 3y = 10$$ and $$4x - y = 5$$. 5. Can systems of equations be solved using matrices or determinants? ### Tip: Always check your solutions by substituting them back into the original equations to ensure they are correct.

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Elimination Method
Substitution Method

Formulas

Linear equations in two variables: ax + by = c

Theorems

Consistency of linear systems

Suitable Grade Level

Grades 8-10

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