Chapter 1
Linear Equations and Graphing
Course: Gre
Subject: Gre Quantitative Reasoning
Chapter: Linear Equations And Graphing
What is a System of Equations?
A system of equations is a set of two or more equations that work together. When we solve a system of equations, we're looking for the point (or points) where all the equations are true at the same time. This point is called the point of intersection and is written as an ordered pair (x, y).
Systems of equations can be written in different forms:
- Slope-intercept form: y = mx + b
- Standard form: Ax + By = C
Types of Solutions
When solving a system of two linear equations, there are three possible outcomes:
| Type of Solution | What It Looks Like | Graphical Representation | | One Solution | The lines intersect at a single point | Two lines crossing | | No Solution | The lines never intersect | Two parallel lines | | Infinite Solutions | Every point works for both equations | The same line (lines overlap) |
Methods for Solving Systems of Equations
There are three main methods for solving systems of equations: graphing, substitution, and elimination. Let's explore each method in detail.
Method 1: Solving by Graphing
When solving by graphing, you plot both lines on the same coordinate plane. The point where they cross is your solution.
Real-World Example: Two Runners
Imagine you and a friend are training for a half-marathon (13 miles). You run at a pace of 1 mile every 9 minutes, while your friend runs 1 mile every 7 minutes. To make it fair, you get a 2-mile head start. Will your friend catch you before the race ends?
Step 1: Write equations for each runner
Your equation: You start at 2 miles and run 1 mile every 9 minutes
- y-intercept (b) = 2
- slope (m) = 1/9 (up 1 mile, over 9 minutes)
- Equation: y = (1/9)x + 2
Friend's equation: Friend starts at 0 miles and runs 1 mile every 7 minutes
- y-intercept (b) = 0
- slope (m) = 1/7 (up 1 mile, over 7 minutes)
- Equation: y = (1/7)x
Step 2: Graph both lines
- Your line starts at (0, 2) and increases slowly
- Friend's line starts at (0, 0) and increases more steeply
- The intersection point shows where and when your friend catches you
Step 3: Interpret the graph From the graph, it appears your friend catches you around 63 minutes at about 9 miles. But to get the exact answer, we need algebra.
Method 2: Solving by Substitution
The substitution method works best when equations are in slope-intercept form (y = mx + b). Since both equations are set equal to y, we can set them equal to each other and solve for x.
Example 1: The Runner Problem (Solved with Substitution)
Equations:
- y = (1/9)x + 2
- y = (1/7)x
Step 1: Set the equations equal to each other (1/9)x + 2 = (1/7)x
Step 2: Solve for x Subtract (1/9)x from both sides: 2 = (1/7)x - (1/9)x
Find a common denominator (63): 2 = (9/63)x - (7/63)x 2 = (2/63)x
Multiply both sides by the reciprocal (63/2): x = 2 × (63/2) = 63 minutes
Step 3: Find y by substituting x back in Using friend's equation: y = (1/7) × 63 = 9 miles
Solution: (63, 9) — Your friend catches you after 63 minutes at the 9-mile mark.
Example 2: Basic Substitution
Solve the system:
- y = 4x
- y = x + 6
Step 1: Set equal: 4x = x + 6 Step 2: Solve for x: 3x = 6 → x = 2 Step 3: Find y: y = 4(2) = 8 Solution: (2, 8)
Example 3: Substitution with Fractions
Solve the system:
- y = (1/2)x + 4
- y = x - 2
Step 1: x - 2 = (1/2)x + 4 Step 2: x - (1/2)x = 4 + 2 (1/2)x = 6 x = 12 Step 3: y = 12 - 2 = 10 Solution: (12, 10)
Example 4: Substitution with Negative Values
Solve the system:
- y = -3x - 15
- y = (1/4)x - 2
Step 1: -3x - 15 = (1/4)x - 2 Step 2: -3x - (1/4)x = -2 + 15 (-12/4 - 1/4)x = 13 (-13/4)x = 13 x = 13 × (-4/13) = -4 Step 3: y = -3(-4) - 15 = 12 - 15 = -3 Solution: (-4, -3)
Example 5: When One Equation is a Constant
Solve the system:
- y = (1/2)x + 3
- y = 5
Step 1: 5 = (1/2)x + 3 Step 2: 2 = (1/2)x x = 4 Step 3: y is already given as 5 Solution: (4, 5)
Method 3: Solving by Elimination
The elimination method (also called addition method) works best when equations are in standard form (Ax + By = C). The goal is to add the equations together so that one variable cancels out.
Key Steps for Elimination:
- Choose which variable to eliminate (x or y)
- Multiply one or both equations so that the coefficients of that variable are opposites
- Add the equations to eliminate that variable
- Solve for the remaining variable
- Substitute back to find the other variable
Example 1: Basic Elimination
Solve the system:
- 3x + 2y = -13
- 3x + 4y = 1
Step 1: Choose to eliminate x (both have 3x) Multiply the first equation by -1: -1(3x + 2y = -13) → -3x - 2y = 13
Step 2: Add the equations -3x - 2y = 13 +3x + 4y = 1 ——————— 0x + 2y = 14
Step 3: Solve for y 2y = 14 → y = 7
Step 4: Find x Substitute y = 7 into either original equation: 3x + 4(7) = 1 3x + 28 = 1 3x = -27 x = -9
Solution: (-9, 7)
Example 2: Elimination with Multiplication
Solve the system:
- -7x - 2y = -13
- x - 2y = 11
Step 1: Choose to eliminate x Multiply the second equation by 7: 7(x - 2y = 11) → 7x - 14y = 77
Step 2: Add the equations -7x - 2y = -13 +7x - 14y = 77 ——————— 0x - 16y = 64
Step 3: Solve for y -16y = 64 → y = -4
Step 4: Find x Substitute y = -4 into x - 2y = 11: x - 2(-4) = 11 x + 8 = 11 x = 3
Solution: (3, -4)
Comparing the Three Methods
| Method | Best Used When | Advantages | Disadvantages | | Graphing | You want a visual estimate | Shows the relationship visually; good for estimating | Less precise; hard to read exact values | | Substitution | Equations are in y = mx + b form | Straightforward; exact answers | Can get messy with fractions | | Elimination | Equations are in Ax + By = C form | Works well with larger numbers; no fractions until the end | Requires careful multiplication |
Advanced Topic: Gaussian Elimination (for 3+ Variables)
For systems with three or more variables, we can use Gaussian elimination, which is a systematic way to eliminate variables step by step until we get a "staircase" form.
Example with Three Variables:
Solve the system:
- x + 2y - z = 3
- 2x - y + 3z = 7
- 3x + y + 2z = 10
Step 1: Eliminate x from equations 2 and 3 using equation 1 Step 2: Eliminate y from the resulting equation Step 3: Solve backwards (find z, then y, then x)
The solution would be found systematically, working from the bottom up.
Key Takeaways
- A system of equations has two or more equations that share variables
- The solution is the point (x, y) that makes ALL equations true
- Systems can have one solution, no solution, or infinitely many solutions
- Substitution works by setting equations equal to each other
- Elimination works by adding equations to cancel variables
- Graphing provides a visual check of your algebraic solution
- The point of intersection represents the moment when two real-world quantities are equal (like runners meeting, costs being the same, etc.)