Solving Systems of Linear Equations: Substitution and Elimination

1. What Are Simultaneous Equations?

Imagine you have a single equation with two variables, like $x+y=10$. How many solutions are there? There are infinitely many! If $x=1$, then $y=9$. If $x=5$, then $y=5$. If $x=20$, then $y=-10$. We can't find a single specific solution.

To find unique values for two variables, we need a second, different equation that also involves those same two variables. For example:

1. $x+y=10$
2. $x-y=2$

This pair of equations is called a system of linear equations, or as it's more commonly known in the UK, a set of simultaneous equations. Solving the system means finding the one pair of values for $x$ and $y$ that makes both equations true.

Graphically, every linear equation represents a straight line. The solution to a system of two linear equations is simply the point where the two lines intersect. This point lies on both lines, so its coordinates ($x,y$) satisfy both equations.

To explore this idea and get a formal introduction, please read the first section of the following guide.

Now that we have the conceptual framework, let's learn the algebraic methods to find this point of intersection precisely without needing to draw a graph.

2. The Elimination Method

The first technique we'll cover is the elimination method (sometimes called the addition method). The strategy is to add or subtract the two equations in a way that eliminates one of the variables, leaving you with a simple one-variable equation that you already know how to solve.

The process is as follows:

1. Align the equations: Write the equations so that the $x$ terms, $y$ terms, and constant terms are aligned in columns.
2. Match coefficients: If necessary, multiply one or both equations by a number so that the coefficient of one of the variables (either $x$ or $y$) is the same in both equations, or are exact opposites (e.g., $+3y$ and $-3y$).
3. Eliminate:
   • If the coefficients are opposites (e.g., $+3y$ and $-3y$), add the two equations.
   • If the coefficients are the same (e.g., $+5x$ and $+5x$), subtract one equation from the other.
4. Solve: Solve the resulting single-variable equation.
5. Substitute back: Substitute the value you just found back into either of the original equations to solve for the other variable.
6. Check your solution: It's good practice to plug both values ($x$ and $y$) into the other original equation to ensure they work.

This video provides a clear walkthrough of the elimination method with several examples.

3. The Substitution Method

The second technique is the substitution method. The strategy here is to rearrange one equation to isolate one variable, and then substitute that expression into the other equation. This also results in a single-variable equation.

The process is as follows:

1. Isolate: Choose one equation and solve for one variable in terms of the other (e.g., rearrange to get $y=\dots$ or $x=\dots$). This is easiest if one variable already has a coefficient of 1 or -1.
2. Substitute: Substitute the expression from step 1 into the other equation. This will leave you with an equation with only one variable.
3. Solve: Solve this new single-variable equation.
4. Substitute back: Substitute the value you just found back into the rearranged equation from step 1 to easily find the second variable.
5. Check: As before, check your solution in the original equations.

Let's return to the mathcentre.ac.uk guide for a detailed explanation and worked examples of this method.

4. Choosing a Method and Further Practice

So, which method is better?

• Use substitution when one of the equations is already solved for a variable, or when one variable has a coefficient of 1 or -1, making it easy to isolate.
• Use elimination when the coefficients of one variable are already the same or opposites, or when all variables have coefficients other than 1, making substitution more cumbersome.

Ultimately, both methods will always work, so you can choose the one you find more comfortable for a given problem.

The guide Simultaneous linear equations contains excellent practice problems. I encourage you to try a few from "Exercises 3" on page 8, as it allows you to choose the method you prefer. The answers are provided so you can check your work.

It's also worth noting that not all systems have a single solution. If the lines are parallel, they never intersect, and there is no solution. Algebraically, you'll end up with a nonsensical statement, like 0 = -102, as shown in the first example in Section 4 of the guide. If the two equations represent the exact same line, there are infinite solutions.

Conclusion

Great work today! You have learned two robust methods for solving systems of linear equations. This is a significant step in your algebraic journey, moving from single-variable to multi-variable problems.

Here are the key takeaways:

• A system of two linear equations can be solved to find a single pair of values $(x,y)$ that satisfies both. This solution represents the point of intersection of the two lines on a graph.
• The elimination method involves adding or subtracting the equations to cancel out one variable.
• The substitution method involves isolating one variable in one equation and substituting its expression into the other equation.

In our next lesson, we will put these new skills to practical use. You will learn how to translate word problems into systems of linear equations and then solve them to find real-world answers.