Sure! Let me create a problem from linear equations and walk through the steps for solving it.

Problem:

Solve the following system of linear equations:

$3x + 4y = 10 \quad \text{(1)}$ $2x - y = 1 \quad \text{(2)}$

Step 1: Choose a method

For this system, I’ll use the substitution method to solve for $x$ and $y$.

Step 2: Solve one equation for one variable

Let’s solve Equation (2) for $y$. From Equation (2): $2x - y = 1$ We can isolate $y$ by adding $y$ to both sides and subtracting $1$ from both sides: $-y = 1 - 2x$ Multiplying both sides by $-1$: $y = 2x - 1$

Step 3: Substitute into the other equation

Now, substitute the expression for $y$ (which is $y = 2x - 1$) into Equation (1): $3x + 4y = 10$ Substituting $y = 2x - 1$: $3x + 4(2x - 1) = 10$

Step 4: Simplify the equation

Distribute the 4 to both terms inside the parentheses: $3x + 8x - 4 = 10$ Combine like terms: $11x - 4 = 10$

Step 5: Solve for $x$

To isolate $x$, add 4 to both sides of the equation: $11x = 14$ Now, divide both sides by 11: $x = \frac{14}{11}$

Step 6: Substitute the value of $x$ into the expression for $y$

Now that we know $x = \frac{14}{11}$, substitute this value into the expression $y = 2x - 1$: $y = 2\left(\frac{14}{11}\right) - 1$ Simplifying: $y = \frac{28}{11} - 1 = \frac{28}{11} - \frac{11}{11} = \frac{17}{11}$

Step 7: Write the final solution

The solution to the system of equations is: $x = \frac{14}{11}, \quad y = \frac{17}{11}$

Conclusion:

The solution to the system of linear equations is $x = \frac{14}{11}$ and $y = \frac{17}{11}$.

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Do you have any questions about this process? Would you like more details on any of the steps?

Here are 5 related questions:

1. How would the elimination method work for this system of equations?
2. What would happen if we had a system with no solution or infinitely many solutions?
3. How do we check the solution we found to ensure it's correct?
4. How can we interpret the solution geometrically using graphs?
5. What is the general method for solving systems of equations with more than two variables?

Tip:

When using the substitution method, always choose to solve the equation that makes isolating one variable easier!