Solving by substitution is a method used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation(s). Here’s a step-by-step guide on how to solve a system of equations by substitution:

Example Problem:

Solve the system of equations: x + y = 7 \tag{1} 2x - y = 4 \tag{2}

Step 1: Solve one equation for one variable

Choose one of the equations to solve for one of the variables. We can solve Equation (1) for $x$ or $y$. Let’s solve for $x$ in terms of $y$:

From Equation (1): $x + y = 7$ Solve for $x$: x = 7 - y \tag{3}

Step 2: Substitute this expression into the other equation

Now, substitute the expression for $x$ from Equation (3) into Equation (2):

Substitute $x = 7 - y$ into $2x - y = 4$: $2(7 - y) - y = 4$ Distribute the 2: $14 - 2y - y = 4$ Simplify: $14 - 3y = 4$ Now, solve for $y$.

Step 3: Solve for the second variable

To solve for $y$: $14 - 3y = 4$ Subtract 14 from both sides: $-3y = 4 - 14$ $-3y = -10$ Divide by -3: $y = \frac{-10}{-3} = \frac{10}{3}$

Step 4: Substitute the value of the second variable back into the first equation

Now that we know $y = \frac{10}{3}$, substitute this value into Equation (3) to solve for $x$:

From Equation (3): $x = 7 - y$ Substitute $y = \frac{10}{3}$: $x = 7 - \frac{10}{3}$ Convert 7 into a fraction: $x = \frac{21}{3} - \frac{10}{3}$ Simplify: $x = \frac{21 - 10}{3} = \frac{11}{3}$

Final Solution:

So, the solution to the system of equations is: $x = \frac{11}{3}, \quad y = \frac{10}{3}$

Summary of Steps:

1. Solve one equation for one variable.
2. Substitute the expression for that variable into the other equation.
3. Solve for the second variable.
4. Substitute that value back into the first equation to find the first variable.

---

Do you need more details or clarifications on any part?

Here are 5 related questions to explore further:

1. What if the system of equations has no solution or infinitely many solutions?
2. Can the substitution method be applied to non-linear systems (e.g., involving quadratics)?
3. How does the elimination method compare to substitution?
4. How do you handle fractions when solving by substitution?
5. What is the best method when the system has larger numbers of variables?

Tip: Always check your solution by substituting the values back into the original equations to ensure they satisfy both equations!