Graphing Systems of Equations

Solving systems by graphing and identifying solutions

Graphing Systems of Equations

What is a System of Equations?

A system is two or more equations with the same variables: {y=2x+1y=−x+4\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}

Solving by Graphing

The solution is the point where the lines intersect.

Steps:

  1. Graph each equation on the same coordinate plane
  2. Find the intersection point
  3. Check the solution in both equations

Types of Solutions

One Solution: Lines intersect at one point

  • Lines have different slopes

No Solution: Lines are parallel

  • Same slope, different y-intercepts
  • Example: y=2x+1y = 2x + 1 and y=2x−3y = 2x - 3

Infinitely Many Solutions: Lines are identical

  • Same slope and same y-intercept
  • Example: y=2x+1y = 2x + 1 and 2y=4x+22y = 4x + 2

📚 Practice Problems

1Problem 1easy

❓ Question:

How many solutions does this system have? {y=3x+2y=3x−5\begin{cases} y = 3x + 2 \\ y = 3x - 5 \end{cases}

💡 Show Solution

Compare the slopes and y-intercepts:

First equation: slope = 3, y-intercept = 2 Second equation: slope = 3, y-intercept = -5

The slopes are equal but the y-intercepts are different.

This means the lines are parallel and never intersect.

Answer: No solution

2Problem 2medium

❓ Question:

Verify that (2,5)(2, 5) is the solution to: {y=2x+1y=−x+7\begin{cases} y = 2x + 1 \\ y = -x + 7 \end{cases}

💡 Show Solution

Substitute x=2x = 2 and y=5y = 5 into both equations:

First equation: y=2x+1y = 2x + 1 5=2(2)+15 = 2(2) + 1 5=4+15 = 4 + 1 5=55 = 5 ✓

Second equation: y=−x+7y = -x + 7 5=−(2)+75 = -(2) + 7 5=−2+75 = -2 + 7 5=55 = 5 ✓

Since (2,5)(2, 5) satisfies both equations, it is the solution.

Answer: Yes, (2,5)(2, 5) is the solution

3Problem 3medium

❓ Question:

Without graphing, determine how many solutions: {y=−2x+32y=−4x+6\begin{cases} y = -2x + 3 \\ 2y = -4x + 6 \end{cases}

💡 Show Solution

Step 1: Convert both to slope-intercept form

First equation is already in the form: y=−2x+3y = -2x + 3

Second equation: 2y=−4x+62y = -4x + 6 y=−2x+3y = -2x + 3

Step 2: Compare Both equations are identical!

When equations are the same, every point on the line is a solution.

Answer: Infinitely many solutions

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