Math Problem Statement

  1. Find the solution of the following systems of equations. a.) . x+y=3 2x-y=0 b.) x-7y=-11 5X+2y=-18

Solution

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

a) System of Equations:

x + y &= 3 \quad \text{(Equation 1)} \\ 2x - y &= 0 \quad \text{(Equation 2)} \end{aligned}$$ **Step 1: Solve one of the equations for one variable.** From Equation (2): $$2x - y = 0 \implies y = 2x$$ **Step 2: Substitute into the other equation.** Substitute $$y = 2x$$ into Equation (1): $$x + 2x = 3 \implies 3x = 3$$ $$x = 1$$ **Step 3: Solve for the second variable.** Substitute $$x = 1$$ into $$y = 2x$$: $$y = 2(1) = 2$$ So, the solution for system **a)** is: $$x = 1, \quad y = 2$$ --- ### b) System of Equations: $$\begin{aligned} x - 7y &= -11 \quad \text{(Equation 3)} \\ 5x + 2y &= -18 \quad \text{(Equation 4)} \end{aligned}$$ **Step 1: Solve one of the equations for one variable.** From Equation (3): $$x = 7y - 11$$ **Step 2: Substitute into the other equation.** Substitute $$x = 7y - 11$$ into Equation (4): $$5(7y - 11) + 2y = -18$$ Expand: $$35y - 55 + 2y = -18 \implies 37y - 55 = -18$$ $$37y = 37 \implies y = 1$$ **Step 3: Solve for the second variable.** Substitute $$y = 1$$ into $$x = 7y - 11$$: $$x = 7(1) - 11 = 7 - 11 = -4$$ So, the solution for system **b)** is: $$x = -4, \quad y = 1$$ --- ### Final Solutions: - For system **a)**: $$x = 1, y = 2$$ - For system **b)**: $$x = -4, y = 1$$ Would you like more details on any of the steps? --- ### 5 Related Questions: 1. How can substitution help solve systems of equations? 2. What other methods can be used to solve systems of equations? 3. Can you explain how elimination works for solving systems of equations? 4. How do you check if a solution to a system of equations is correct? 5. How can systems of equations be applied in real-world problems? **Tip:** When solving systems of equations, substitution is effective if one variable is already isolated or easy to isolate!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
System of Equations
Substitution Method

Formulas

Solving systems of linear equations using substitution
Basic algebraic manipulation

Theorems

Substitution Method for solving systems of equations

Suitable Grade Level

Grades 8-10

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