Solve System of Linear Equations: 3x + y = 3 and 7x

Math Problem Statement

\begin{aligned} 3x+y&=3 \\ 7x-y&=2 \end{aligned}

You are dealing with a system of linear equations:

3x + y &= 3 \hspace{0.5cm} \text{(Equation 1)} \\ 7x - y &= 2 \hspace{0.5cm} \text{(Equation 2)} \end{aligned}$$ To solve this system, we can use the **method of elimination**. Let's add the two equations together to eliminate $$y$$: $$(3x + y) + (7x - y) = 3 + 2$$ This simplifies to: $$10x = 5$$ Now, solve for $$x$$: $$x = \frac{5}{10} = \frac{1}{2}$$ Now that we know $$x = \frac{1}{2}$$, substitute this value into one of the original equations to solve for $$y$$. Let's use Equation 1: $$3\left(\frac{1}{2}\right) + y = 3$$ This simplifies to: $$\frac{3}{2} + y = 3$$ Now, subtract $$\frac{3}{2}$$ from both sides: $$y = 3 - \frac{3}{2} = \frac{6}{2} - \frac{3}{2} = \frac{3}{2}$$ So, the solution to the system is: $$x = \frac{1}{2}, \quad y = \frac{3}{2}$$ Would you like more details on any specific step, or do you have any further questions? Here are 5 related questions to deepen your understanding: 1. How can you check if the solution $$\left( \frac{1}{2}, \frac{3}{2} \right)$$ is correct? 2. How would you solve this system using substitution instead of elimination? 3. Can you write the system of equations in matrix form and solve it? 4. What happens if both equations represent the same line? 5. How would the process change if the equations involved fractions or decimals? **Tip:** When using elimination, it's often helpful to multiply one or both equations by constants to align variables for elimination.

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Algebra
Linear Equations
Systems of Equations
Elimination Method

Addition of equations for elimination
Substitution for solving systems of equations

Elimination Method

Grades 8-10

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