5.1 Solve Systems of Equations by Graphing

Solving Systems of Equations by Graphing

A system of linear equations is two or more equations with the same variables. Solving the system means finding the values of those variables that make all the equations true at once. Graphing is one way to do this: you plot each equation as a line, and the point where the lines cross is your solution.

The Coordinate Plane

The Cartesian coordinate plane is the grid you'll graph on. It has a horizontal x-axis and a vertical y-axis that cross at the origin $(0, 0)$. Every point on the plane is written as an ordered pair $(x, y)$, where $x$ tells you how far left or right and $y$ tells you how far up or down.

The axes divide the plane into four quadrants, numbered I through IV counterclockwise starting from the upper right. You won't always need to think about quadrants when solving systems, but knowing where your points fall helps you graph accurately.

Verifying a Point as a Solution

Before you graph anything, you should know how to check whether a given point is a solution to a system. The idea is simple: plug the point's $x$ and $y$ values into each equation and see if both equations balance.

1. Take the ordered pair $(x, y)$ and substitute $x$ and $y$ into the first equation. Simplify both sides. Do they equal each other?
2. Do the same for the second equation.
3. If the point makes both equations true, it's a solution to the system. If it fails even one equation, it's not.

Graphing to Find the Intersection

This is the core skill of the section. You graph both equations on the same coordinate plane and look for where the lines cross.

1. Put each equation in a form that's easy to graph. Slope-intercept form ($y = mx + b$) is usually the most convenient because you can read the slope and y-intercept directly.
2. Find at least two points for each line. You can plug in values of $x$ and solve for $y$, or use the y-intercept as one point and the slope to find a second. Using three points is even better because the third point acts as a check.
3. Plot the points and draw each line. Use a ruler so your lines are straight and accurate. Extend the lines across the graph.
4. Identify the intersection point. Read the $x$ and $y$ coordinates where the two lines cross. This ordered pair is the solution to the system.
5. Check your answer. Substitute the intersection point back into both original equations to verify it works (use the verification steps above).

A common mistake is reading the intersection point wrong because the lines were drawn sloppily. Take your time with the graphing, and always verify by plugging the point back in.

Three Types of Solutions

Not every system has a single neat intersection point. The graph tells you which type you're dealing with:

• One solution (consistent and independent): The two lines cross at exactly one point. This happens when the lines have different slopes. Most systems you'll encounter fall into this category.
• No solution (inconsistent): The two lines are parallel, meaning they never cross. Parallel lines have the same slope but different y-intercepts. For example, $y = 2x + 1$ and $y = 2x - 3$ will never intersect.
• Infinitely many solutions (consistent and dependent): The two lines are actually the same line, sitting right on top of each other. They have the same slope and the same y-intercept. Every point on the line satisfies both equations. For example, $y = 2x + 1$ and $2y = 4x + 2$ are the same line written differently.

Real-World Applications

Word problems that use systems of equations follow a predictable pattern:

1. Identify the unknowns and assign them to variables. For instance, let $x$ = the number of adult tickets and $y$ = the number of child tickets.
2. Write two equations from the information given. Each equation captures a different relationship. For example, one equation might represent the total number of tickets sold, and the other might represent the total revenue.
3. Graph both equations on the same coordinate plane and find the intersection point.
4. Interpret the solution in context. The intersection $(30, 20)$ might mean 30 adult tickets and 20 child tickets were sold. Always state what the numbers mean, not just what they are.

Graphing works well for getting an approximate answer or for visualizing the problem, but keep in mind that reading exact coordinates from a graph can be tricky when the solution involves fractions or decimals. In later sections, you'll learn algebraic methods (substitution and elimination) that handle those cases more precisely.