GED Math: Systems of Equations

Systems of equations model situations with two unknown quantities and two pieces of information. Steps: (1) define two variables (say x and y) with units; (2) write one equation from each piece of information, often a total count and a total value; (3) solve by substitution or elimination; (4) answer the exact question that was asked, with units, and check against the story. A typical pair is: (number of items) x + y = total, and (price × number) 8x + 5y = total cost.

1. What Is a System of Equations? Checking a Solution

A system of linear equations is two (or more) linear equations that use the same variables. Each equation is a straight line, so solving the system means finding the values of x and y that make both equations true at once. That solution is the point where the two lines intersect, written as an ordered pair (x, y). There are several ways to solve a system: graphing (draw both lines and read the intersection), substitution (solve one equation for a variable and put it into the other), and elimination (add or subtract equations to cancel a variable). Whatever method you use, you can always check an answer by substituting it into both original equations. Test tip: when a question asks whether a point is a solution, you do not need to solve; just test the point in both equations.

2. Solving a System by Graphing

To solve by graphing: (1) make a T-chart for each equation by finding two ordered pairs (use x = 0 to find y, then y = 0 to find x, or any easy values); (2) plot the points and draw both lines on the same grid; (3) find the point of intersection and write it as (x, y). Always check by substituting that point into both equations, because reading a graph can be off by a small amount. Graphing is a good method when the answer is a whole-number point, and when a question already shows the graph, simply read the crossing point.

3. Solving a System by Substitution

Substitution works best when one equation already has a variable alone, or can easily be solved for one. Steps: (1) solve one equation for y (or x); (2) substitute that expression into the other equation, which leaves one equation with one variable; (3) solve for that variable; (4) substitute the value back into either equation to find the second variable; (5) write the solution as (x, y) and check it in both equations. Be careful to put parentheses around the substituted expression so negative signs are distributed correctly. Test tip: make sure you substitute into the other equation; substituting back into the same equation just gives 0 = 0 or a true statement with no information.

4. Solving a System by Elimination

Elimination is a third method that often works faster when both equations are written as ax + by = c. The idea is to add or subtract the equations so that one variable cancels. Steps: (1) line the equations up so the x-terms, y-terms and constants are in columns; (2) if no variable has matching or opposite coefficients, multiply one or both equations by a number to create them; (3) add the equations if the coefficients are opposites (like +3y and −3y), or subtract if they are the same; (4) solve for the remaining variable; (5) substitute back to find the other variable and check in both equations.

5. Special Systems: Parallel, Same Line, and Intersecting Lines

Not every system has exactly one solution. Rewrite both equations in slope-intercept form y = mx + b and compare. If the slopes are different, the lines cross exactly once: one solution. If the slopes are the same but the y-intercepts differ, the lines are parallel and never meet: no solution (algebra produces a false statement like 0 = 5). If the slopes and the y-intercepts are both the same, the equations describe the same line: infinitely many solutions (algebra produces a true statement like 0 = 0). This lets you answer “does this line intersect that line?” without graphing: a line with a different slope always intersects.

6. Systems in Word Problems

Systems of equations model situations with two unknown quantities and two pieces of information. Steps: (1) define two variables (say x and y) with units; (2) write one equation from each piece of information, often a total count and a total value; (3) solve by substitution or elimination; (4) answer the exact question that was asked, with units, and check against the story. A typical pair is: (number of items) x + y = total, and (price × number) 8x + 5y = total cost.

Common Mistakes to Avoid

• Finding only x and forgetting to find y. A solution is an ordered pair.
• Writing the pair as (y, x) instead of (x, y).
• Substituting into the same equation you solved instead of the other one.
• Forgetting parentheses when substituting, so a minus sign is not distributed: 5x + (4 − 2x) is not 5x + 4 − 2x unless you keep the sign correct.
• When eliminating, adding equations when the coefficients are the same (you should subtract) or subtracting only part of an equation.
• Multiplying only one side of an equation when scaling it.
• Checking a point in only one equation. It must satisfy both.
• Assuming every system has one solution. Same slope can mean no solution or infinitely many.

Frequently Asked Questions