5.2 Solving Systems of Equations by Substitution

Solving Systems of Equations by Substitution

Solving a system of equations means finding the values of $x$ and $y$ that make both equations true at the same time. The substitution method works by isolating one variable in one equation, then plugging that expression into the other equation. This reduces the system down to a single equation with one variable, which you already know how to solve.

This method is especially handy when one equation is already solved for a variable (like $y = 3x - 2$) or when a variable has a coefficient of 1 or -1, making it easy to isolate.

Understanding Systems of Equations

An equation is a mathematical statement that two expressions are equal. A system of equations (also called simultaneous equations) is two or more equations considered together. The solution to a system is the ordered pair $(x, y)$ that satisfies every equation in the system.

For example, the system $2x + y = 7$ and $x - y = 2$ has the solution $(3, 1)$ because plugging in $x = 3$ and $y = 1$ makes both equations true.

The Substitution Method: Step-by-Step

Here's the process:

1. Isolate one variable in whichever equation makes it easiest. Pick the variable that has a coefficient of 1 or -1 if possible.
2. Substitute that expression into the other equation, replacing the variable entirely.
3. Solve the resulting single-variable equation.
4. Back-substitute the value you found into the expression from Step 1 to get the other variable.
5. Check your solution by plugging both values into the original equations to confirm they work.

Worked Example:

Solve the system: $2x + 3y = 12$ and $x - y = 1$

1. The second equation is easiest to rearrange. Isolate $x$: $x = y + 1$

2. Substitute $y + 1$ for $x$ in the first equation: $2(y + 1) + 3y = 12$

3. Solve for $y$:

   • $2y + 2 + 3y = 12$
   • $5y + 2 = 12$
   • $5y = 10$
   • $y = 2$

4. Back-substitute $y = 2$ into $x = y + 1$: $x = 2 + 1 = 3$

5. Check: Does $(3, 2)$ work in both originals?

   • $2(3) + 3(2) = 6 + 6 = 12$ ✓
   • $3 - 2 = 1$ ✓

The solution is $(3, 2)$.

When Substitution Is the Best Choice

Not every method works equally well for every system. Substitution tends to be the most efficient in these situations:

• One equation is already solved for a variable. If you're given $y = 2x + 1$, you can skip straight to Step 2. No rearranging needed.
• A variable has a coefficient of 1 or -1. Equations like $x + 4y = 9$ or $-y + 3x = 5$ let you isolate a variable without introducing fractions.
• The other methods would create messy arithmetic. If both equations have fractional coefficients like $\frac{2}{3}x + \frac{1}{2}y = 5$, elimination often requires multiplying by large numbers. Substitution can sometimes keep things simpler.

If both variables in both equations have large coefficients and neither is easy to isolate, the elimination method might be a better fit.

Real-World Applications

Many word problems translate naturally into systems of equations. Here's how to approach them with substitution:

1. Define your variables. Clearly state what $x$ and $y$ represent (e.g., let $x$ = number of adult tickets, $y$ = number of student tickets).
2. Write two equations from the information given. You need two independent relationships to solve for two unknowns.
3. Solve using substitution following the steps above.
4. Interpret your answer in context. Make sure the values make sense for the problem. If you get $x = -3$ for a number of tickets, something went wrong.

Quick Example: A store sold apples and oranges for a total of 5 fruits. Apples cost $2 each and oranges cost $3 each, and the total was $12. Setting up: $x + y = 5$ and $2x + 3y = 12$. From the first equation, $x = 5 - y$. Substituting gives $2(5 - y) + 3y = 12$, which simplifies to $y = 2$ and $x = 3$. The store sold 3 apples and 2 oranges.