Math Problem Statement

solve

Solve each system of equations. a) { 2x - 4y = 10, x + 5y = 40 } b) { 3x - 5y = 4, -2x + 6y = 18 }

Solution

Let's solve each system of equations step-by-step.

Problem (a)

The system is:

2x - 4y = 10 \\ x + 5y = 40 \end{cases}$$ 1. **Solve for $$x$$ from the second equation**: $$x + 5y = 40 \implies x = 40 - 5y$$ 2. **Substitute $$x = 40 - 5y$$ into the first equation**: $$2(40 - 5y) - 4y = 10$$ Expanding and simplifying: $$80 - 10y - 4y = 10$$ $$80 - 14y = 10$$ $$-14y = -70$$ $$y = 5$$ 3. **Substitute $$y = 5$$ back into $$x = 40 - 5y$$**: $$x = 40 - 5(5) = 40 - 25 = 15$$ **Solution for (a):** $$(x, y) = (15, 5)$$ --- ### Problem (b) The system is: $$\begin{cases} 3x - 5y = 4 \\ -2x + 6y = 18 \end{cases}$$ 1. **Multiply the first equation by 2 and the second equation by 3 to align the coefficients of $$x$$**: $$\begin{cases} 6x - 10y = 8 \\ -6x + 18y = 54 \end{cases}$$ 2. **Add the equations to eliminate $$x$$**: $$(6x - 10y) + (-6x + 18y) = 8 + 54$$ $$8y = 62$$ $$y = \frac{62}{8} = \frac{31}{4}$$ 3. **Substitute $$y = \frac{31}{4}$$ back into the first equation to find $$x$$**: $$3x - 5\left(\frac{31}{4}\right) = 4$$ $$3x - \frac{155}{4} = 4$$ $$3x = 4 + \frac{155}{4} = \frac{16 + 155}{4} = \frac{171}{4}$$ $$x = \frac{171}{4} \div 3 = \frac{171}{12} = \frac{57}{4}$$ **Solution for (b):** $$\left( x, y \right) = \left( \frac{57}{4}, \frac{31}{4} \right)$$ --- ### Final Answers: - (a) $$(15, 5)$$ - (b) $$\left( \frac{57}{4}, \frac{31}{4} \right)$$ Would you like more details on any part of these solutions? --- Here are some additional questions to explore: 1. How do you know when to add or subtract equations when solving a system? 2. What are other methods for solving systems of equations (besides substitution and elimination)? 3. How can you verify the solution of a system of equations? 4. What happens if a system of equations has no solutions? 5. How can graphing help understand solutions to systems of equations? **Tip:** When solving a system, check if you can simplify first by aligning coefficients for easier elimination.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Substitution Method
Elimination Method

Formulas

Substitution: Solve for one variable in terms of the other.
Elimination: Align and add/subtract equations to eliminate a variable.

Theorems

Properties of Equality
Linear Combination of Equations

Suitable Grade Level

Grades 9-10

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