LESSON 21

SYSTEMS OF EQUATIONS

Finding Where Two Lines Meet

Two Equations, One Solution

A system of equations is a set of two or more equations that share the same variables. The solution to a system is the set of values that satisfies all equations at the same time. Graphically, this is the point where the lines intersect — the one coordinate pair that lies on both lines simultaneously.

Systems of equations are used to solve problems that involve two unknown quantities with two conditions — mixing solutions, comparing pricing plans, finding break-even points in business, or determining when two moving objects meet. There are three methods for solving them: graphing, substitution, and elimination.

Solving by Graphing

To solve a system by graphing, graph both equations on the same coordinate plane and identify the point of intersection. If the lines cross at one point, there is exactly one solution. If the lines are parallel (same slope, different y-intercepts), there is no solution — the system is inconsistent. If the lines are identical (same slope and same y-intercept), there are infinitely many solutions — the system is dependent.

Example:y = x + 1 and y = −x + 5 Graph both lines → they intersect at (2, 3) Solution: x = 2, y = 3 Check: 3 = 2 + 1 ✓ 3 = −2 + 5 ✓

Solving by Substitution

Substitution works best when one equation is already solved for one variable. Step 1: isolate one variable in one equation. Step 2: substitute that expression into the other equation. Step 3: solve for the remaining variable. Step 4: substitute back to find the first variable. Step 5: check the solution in both original equations.

Example:y = 2x − 1 and 3x + y = 9 Substitute y: 3x + (2x − 1) = 9 5x − 1 = 9 → 5x = 10 → x = 2 y = 2(2) − 1 = 3 Solution: (2, 3)

Solving by Elimination

Elimination (also called addition) works by adding or subtracting the two equations to cancel out one variable. If the coefficients of one variable are opposites, add the equations directly. If not, multiply one or both equations by a constant first to create opposite coefficients. Then solve for the remaining variable and substitute back.

Example:2x + 3y = 12 and 4x − 3y = 6 Add equations: 6x = 18 → x = 3 Substitute: 2(3) + 3y = 12 → 3y = 6 → y = 2 Solution: (3, 2) Check: 2(3)+3(2)=12 ✓ 4(3)−3(2)=6 ✓

Special Cases and Word Problems

When solving a system yields a true statement like 0 = 0, the system has infinitely many solutions (the equations describe the same line). When it yields a false statement like 0 = 5, the system has no solution (the lines are parallel). For word problems, define variables for the two unknowns, write one equation for each condition given, then solve the system using any method.

Example:Two numbers add to 20 and their difference is 4. Let x + y = 20 and x − y = 4 Add: 2x = 24 → x = 12 y = 20 − 12 = 8 The two numbers are 12 and 8.

Key Facts

Systems of equations date back to ancient China. The text 'The Nine Chapters on the Mathematical Art' (circa 200 BCE) solved systems of linear equations using a method equivalent to modern Gaussian elimination — nearly 2,000 years before Gauss.

The break-even point in business — where revenue equals cost — is found by solving a system of two linear equations. Every entrepreneur who calculates when their business becomes profitable is using this concept.

A system of two linear equations in two variables can have exactly 0, 1, or infinitely many solutions — never 2 or more finite solutions. Two distinct lines either miss each other, cross once, or are the same line.

Modern computers solve systems with millions of equations and variables using matrix methods derived from elimination. This underlies everything from weather forecasting to Google's PageRank algorithm.

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