Chapter 4: Simultaneous Linear Equations

Section 4.1: Understanding Solutions of Simultaneous Linear Equations

In this section, you will analyze and solve pairs of simultaneous linear equations — two equations that must be true at the same time. You'll also learn that the solutions to a system (a pair) of two linear equations with two variables match up with the point where their graphs cross. That's because a point where two graphs cross is the one place that makes both equations true at once.

Simultaneous linear equations are a pair of equations that look like $Ax + By = C$, where $A$, $B$, and $C$ are specific numbers (they can be positive or negative). When we call two equations "simultaneous," we're asking: are there values for $x$ and $y$ that make both equations true at the same time? If we can find such values, that pair of numbers is called a solution.

For example, look at this pair of equations: $x + 2y = 10$ and $x - 3y = 0$. These describe two different relationships between $x$ and $y$. If we need both to be true together, they're simultaneous equations. One solution is $x = 6, y = 2$, because plugging those numbers in makes both equations true. In this chapter, we'll explore both algebraic and graphical methods for finding solutions to simultaneous linear equations.

Example 1

Suppose $x = 21$ and $y = 17$. This is technically a pair of simultaneous linear equations, and clearly it has only one solution: $x = 21, y = 17$.

Now imagine I want to turn this into a puzzle. Let's say these numbers are the ages of my older siblings, and someone asks me to figure them out. I could say, "The sum of their ages is 38." But that's not enough — many pairs of numbers add up to 38. So I add another clue: "The difference in their ages is 4."

Now can the puzzle-solver find their ages? If they call the ages $x$ and $y$, they can write down both clues as equations. "The sum of their ages is 38" becomes $x + y = 38$. "The difference in their ages is 4" becomes $x - y = 4$. Together, these two equations

$$x + y = 38$$
$$x - y = 4$$

form a pair of simultaneous linear equations, and the actual ages of my siblings are the solution. But is this enough information to solve it? If we know the sum and the difference of two numbers, can we find the original numbers? The answer is yes — and to solve the problem, we need to figure out how to work backward from the sum and difference to the original numbers.

Example 2

Here's another example, presented as a game, that shows the same idea:

1. Pick two numbers.
2. Double one number and add the other. Tell me the result.
3. Now switch the numbers and repeat the same steps. Tell me that result.

Say the numbers I receive are 30 and 27. After just a second, I announce, "Your two numbers were 8 and 11."

How did I solve it so quickly? Let's figure out the process by working through the math from Examples 1 and 2. In both cases, we start with a specific pair of numbers, then perform algebraic operations on them. In the first example, we added the numbers together and then subtracted them: from $x = 21, y = 17$, we created $x + y = 38$ and $x - y = 4$. The challenge is to reverse this process.

One idea: add the two equations together.

$$x + y = 38$$
$$x - y = 4$$

Adding gives us:

$$(x + y) + (x - y) = 38 + 4$$

which simplifies to $2x = 42$, so $x = 21$.

Now let's subtract the two equations instead:

$$x + y = 38$$
$$x - y = 4$$

Subtracting gives us:

$$(x + y) - (x - y) = 38 - 4$$

which simplifies to $2y = 34$, so $y = 17$.

Now let's use the same method on the second example. Call the two chosen numbers $x$ and $y$. In step 2, we calculate $2x + y$ and get the result 30. In step 3, we calculate $2y + x$ and get 27. So we now know:

$$2x + y = 30$$
$$2y + x = 27$$

Let's add these:

$$(2x + y) + (2y + x) = 57$$

which simplifies to $3(x + y) = 57$, or $x + y = 19$.

Now let's subtract:

$$(2x + y) - (2y + x) = 30 - 27$$

which gives $x - y = 3$. Now that we know both the sum and the difference of the two numbers, we can use the same technique as Example 1 to find $x = 11$ and $y = 8$.

The point of these examples is to show how we can transform one pair of simultaneous equations into another pair — without changing the solution. When we're trying to disguise the numbers, we keep applying operations until the information is well-scrambled. When we're trying to solve the puzzle, we choose operations that unscramble it. The tools we use are:

a. Add equal amounts to both sides of an equation (this is how $x = 21, y = 17$ became $x + y = 38$).

b. Subtract equal amounts from both sides (this is how $x = 21, y = 17$ became $x - y = 4$).

c. Multiply both sides by the same nonzero number (for example, when we had $2y = 34$, we multiplied both sides by $\frac{1}{2}$).

Using these operations, we can transform one pair of simultaneous equations into a different pair — while keeping the same solution. Notice that since we have two unknowns, we always need two equations at every step. If I tell you only that the sum of two numbers is 38, that's not enough information to find them. I'd need to either tell you one of the numbers directly, or give you another clue — like the difference between them.

We don't use these operations randomly — we choose them carefully to reach our goal. As we'll see in the next section, the numbers and the form of the equations themselves guide us toward the right operations to use. There's one more tool for solving equations: substitution.

d. Replace an expression in one equation with an equal expression taken from the other equation.

Example 3

Lovasz has 5 marbles more than twice the number of marbles Tonio has. Together, they have 107 marbles. How many marbles does Lovasz have?

Solution. First, let's represent the unknowns with letters. Let $L$ stand for the number of marbles Lovasz has, and $T$ stand for the number Tonio has. We're told "Lovasz has 5 marbles more than twice Tonio's," which translates to $L = 5 + 2T$. We're also told the total is 107 marbles, so $L + T = 107$.

Since the first equation tells us that $L$ and $5 + 2T$ represent the same value, we can substitute $5 + 2T$ in place of $L$ in the second equation:

$$5 + 2T + T = 107$$

From Chapter 1, we know how to solve a linear equation with one variable: combine like terms on the left, then subtract 5 from both sides to get $3T = 102$, so $T = 34$ — Tonio has 34 marbles.

To find $L$, we go back to the first equation and substitute 34 for $T$: $L = 5 + 2(34)$, so $L = 73$. We can double-check using the second equation: $73 + 34 = 107$. ✓

Before we dive deeper into using these operations to solve systems of equations, let's look at how this process appears on a graph.

Example 4

Consider the linear equations $3x + y = 7$ and $x + 3y = 5$. Graph both equations on a coordinate plane, and find the coordinates $(x, y)$ where they cross.

Solution. The slope of the first line is $-3$, and the slope of the second is $-\frac{1}{3}$. Since the slopes are different, the lines aren't parallel, so they must cross somewhere. Using a point on each line — for example, the y-intercepts $(0, 7)$ for the first line and $(0, \frac{5}{3})$ for the second — we can graph the lines, as shown in Figure 1.

The graph shows that the lines intersect at $(2, 1)$. Let's confirm this by checking that $x = 2, y = 1$ makes both equations true:

$$3(2) + 1 = 7$$
$$2 + 3(1) = 5$$

Both check out! In this example, we graphed the two linear equations and read the coordinates $(2, 1)$ directly from the graph. Since this point lies on both lines, $x = 2, y = 1$ satisfies both equations. This is exactly what it means to "solve a pair of simultaneous equations."

Example 5

Consider the linear equations $x - 2y = 8$ and $2x + 5y = 34$. Graph each equation on the same set of axes, and read off the coordinates $(x, y)$ where they intersect.

Solution. Once again, the two lines have different slopes ($\frac{1}{2}$ and $-\frac{2}{5}$), so they aren't parallel and must cross at some point. Graphing these equations (see Figure 2), we find the point of intersection at $(12, 2)$. Since $(12, 2)$ lies on both lines, $x = 12, y = 2$ satisfies both equations.

Since two non-parallel lines only cross at one point, this is the only solution — it's unique.

But what happens when the lines are parallel? Let's look at that case next.

Example 6

Consider the linear equations $2x + 5y = 10$ and $4x + 10y = 40$. Graph each equation and look for a point where they cross.

These two equations produce two different lines (they have different y-intercepts, $(0, 2)$ and $(0, 4)$), but the lines are parallel — they have the same slope ($-\frac{2}{5}$). Since the lines never meet, there is no point of intersection. This tells us there are no values of $x$ and $y$ that satisfy both equations at once. The graph in Figure 3 confirms this.

Example 7

Now consider the linear equations $2x + 5y = 20$ and $4x + 10y = 40$. These two equations actually produce the same graph, because the lines have identical slopes and identical y-intercepts.

If we rewrite both equations in slope-intercept form, they both simplify to $y = -\frac{2}{5}x + 4$. In this case, there are infinitely many solutions, since every point on the line satisfies both equations. Notice that one of the original equations is simply a multiple of the other (divide both sides of the second equation by 2 to see this) — meaning they're equivalent expressions describing the exact same relationship. This pattern always shows up whenever a pair of simultaneous linear equations has more than one solution.

Summary: Three Possibilities

Given a pair of linear equations, there are three possible outcomes when looking for simultaneous solutions:

1. Different slopes: If the rate of change of $y$ with respect to $x$ (the slope) is different for the two equations, the lines are not parallel, and they intersect at exactly one point. That point's coordinates give the unique solution to the pair of equations.

1. Same slope, different intercepts: If the slope is the same for both equations but they have different y-intercepts, the lines are parallel but different. In this case, there is no solution to the simultaneous equations.

1. Same slope, same intercept: If the slope and y-intercept are the same for both equations, they describe the exact same line. In this case, every point on the line is a solution — there are infinitely many.

In short: if two lines have different slopes (they're not parallel), there's exactly one point where they cross. If two lines have the same slope (they're parallel), then either there's no solution at all, or they're actually the same line, giving many solutions.

Let's take a look at what happens graphically when we apply these operations to simultaneous equations. Start with the equations from Example 5: $x - 2y = 8$ and $2x + 5y = 34$. Their graphs are shown in Figure 2. If we add the two equations together, we get $3x + 3y = 42$. If we subtract the first from the second, we get $x + 7y = 26$. Adding the graphs of these two new equations to Figure 2 gives us Figure 4. The green and purple lines