Lesson 2: Systems of Linear Equations

New Concept

A system of linear equations involves two or more equations with the same variables. We'll learn to find the single ordered pair $(x, y)$ that solves both, often by graphing to find where the lines intersect.

What’s next

Next, you'll work through interactive examples to practice solving systems by graphing and tackle challenge problems based on real-world scenarios.

Property

A pair of linear equations with the same variables is called a system of linear equations. When we consider two equations together, we often use the same variables for both equations, like this:
$y = 6x + 1000$
$y = 2x + 1200$

Examples

• A gym offers two plans. Plan A: $y = 20x + 50$. Plan B: $y = 30x + 20$. This is a system where $y$ is the total cost for $x$ months.
• Two competing internet providers have different pricing. Company 1: $C = 10t + 100$. Company 2: $C = 15t + 50$. This system compares the cost $C$ over $t$ months.
• A student is saving money. Account 1: $A = 5w + 40$. Account 2: $A = 10w + 10$. This system models the amount $A$ in each account after $w$ weeks.

Explanation

Think of a system as two related math stories (equations) that use the same characters (variables). We analyze them together to find a shared outcome or a point where both are true simultaneously.

Property

A solution to a system of equations is an ordered pair $(x, y)$ that satisfies each equation in the system. To check whether an ordered pair is a solution, substitute the coordinates into each equation to verify that they result in true statements.

Examples

• Is $(2, 5)$ a solution to the system $y = x + 3$ and $y = 2x + 1$? For the first equation, $5 = 2 + 3$ is true. For the second, $5 = 2(2) + 1$ is true. Yes, it is the solution.
• Is $(1, 4)$ a solution to the system $y = 3x + 1$ and $y = -x + 3$? For the first, $4 = 3(1) + 1$ is true. For the second, $4 = -(1) + 3$ or $4 = 2$ is false. No, it is not a solution.
• To verify that $(3, 7)$ solves the system $y = 2x + 1$ and $y = 4x - 5$, we check both. $7 = 2(3) + 1$ becomes $7 = 7$ (True). $7 = 4(3) - 5$ becomes $7 = 7$ (True). So it is a solution.

Explanation

The solution is the one special ordered pair that works in both equations of the system. It is the single point that lies on both lines when you graph them, representing the value where they are equal.

Property

The solution of a system satisfies both equations in the system, so the point that represents the solution must lie on both graphs. It is the intersection point of the two lines described by the system. Thus, we can solve a system of equations by graphing the equations and looking for the point where the graphs intersect.

Examples

• To solve the system $y = x + 2$ and $y = -x + 4$, graph both lines. They intersect at the point $(1, 3)$. Therefore, the solution is $(1, 3)$.
• Consider the system $y = 2x$ and $y = 3x - 1$. Graphing these lines reveals they cross at $(1, 2)$. This ordered pair is the solution to the system.
• For the system $y = -2x + 5$ and $y = x - 1$, plotting both lines shows an intersection at $(2, 1)$. This means $x=2$ and $y=1$ is the solution.

Explanation

Imagine each equation is a path. The solution to the system is the treasure buried at the exact spot where the two paths cross. The coordinates of this intersection point will solve both equations.

Property

1 Consistent and independent system. The graphs of the two lines intersect in exactly one point. The system has exactly one solution.

2 Inconsistent system. The graphs of the equations are parallel lines and hence do not intersect. An inconsistent system has no solutions.

3 Dependent system. All the solutions of one equation are also solutions to the second equation, and hence are solutions of the system. The graphs of the two equations are the same line. A dependent system has infinitely many solutions.

Property

Many practical problems involve two or more unknown quantities. Often it is easier to solve these problems by using two variables and writing a system of equations.

1. Assign variables to the two unknown quantities.
2. Write two equations that model the problem.
3. Solve the system (e.g., by graphing).
4. Interpret the solution in the context of the problem.

Examples

• A farm has chickens and pigs. There are 20 animals total and 50 legs total. Let $c$ be chickens and $p$ be pigs. The system is $c + p = 20$ and $2c + 4p = 50$. The solution is $p=5, c=15$.
• You buy 10 items, some apples ($a$) and some bananas ($b$). Apples cost 1 dollar each and bananas cost 2 dollars each, for a total of 13 dollars. The system is $a+b=10$ and $1a+2b=13$. The solution is $b=3, a=7$.
• The perimeter of a rectangle is 30 cm. Its length ($l$) is 5 cm more than its width ($w$). The system is $2l + 2w = 30$ and $l = w + 5$. The solution is $w=5$ cm and $l=10$ cm.

Explanation

When a word problem gives you two unknowns and two clues, you can create a system of two equations. Solving this system helps you find the value of both unknown quantities at once.