Graphing Systems of Equations

Introduction to Graphing Systems

Graphing is a visual method for solving systems of equations. The solution is the point (or points) where the graphs intersect.

Advantages:

• Visual understanding of the solution
• Can see if there are no solutions or infinite solutions
• Helps understand relationship between equations

Disadvantages:

• May be imprecise (especially with non-integer solutions)
• Time-consuming without technology
• Requires accurate graphing

Understanding Solutions Graphically

For a system of two linear equations:

One Solution (Consistent and Independent):

• The lines intersect at exactly ONE point
• Different slopes
• The intersection point (x, y) is the solution

No Solution (Inconsistent):

• The lines are parallel (never meet)
• Same slope, different y-intercepts
• No common point

Infinitely Many Solutions (Consistent and Dependent):

• The lines are identical (overlap completely)
• Same slope, same y-intercept
• Every point on the line is a solution

Steps for Solving by Graphing

Step 1: Write both equations in slope-intercept form (y = mx + b)

Step 2: Graph the first line

• Plot the y-intercept (0, b)
• Use the slope to find more points
• Draw the line

Step 3: Graph the second line on the same axes

• Plot its y-intercept
• Use its slope to find more points
• Draw the line

Step 4: Find the intersection point

• Identify where the lines cross
• Read the coordinates (x, y)

Step 5: Check the solution

• Substitute into both original equations
• Verify it works in both

Detailed Example 1: Lines That Intersect

Solve by graphing: y = 2x - 1 y = -x + 5

Step 1: Both already in slope-intercept form

Step 2: Graph y = 2x - 1

• Y-intercept: (0, -1)
• Slope: 2 = 2/1 (up 2, right 1)
• Points: (0, -1), (1, 1), (2, 3)
• Draw line through points

Step 3: Graph y = -x + 5

• Y-intercept: (0, 5)
• Slope: -1 = -1/1 (down 1, right 1)
• Points: (0, 5), (1, 4), (2, 3)
• Draw line through points

Step 4: Intersection point: (2, 3)

Step 5: Check

• y = 2(2) - 1 = 3 ✓
• y = -(2) + 5 = 3 ✓

Solution: (2, 3)

Detailed Example 2: Converting to Slope-Intercept Form

Solve by graphing: 2x + y = 8 x - y = 1

Step 1: Convert to slope-intercept form

First equation: y = -2x + 8 (slope = -2, y-intercept = 8)

Second equation: -y = -x + 1 y = x - 1 (slope = 1, y-intercept = -1)

Step 2: Graph y = -2x + 8

• Points: (0, 8), (1, 6), (2, 4), (3, 2), (4, 0)

Step 3: Graph y = x - 1

• Points: (0, -1), (1, 0), (2, 1), (3, 2), (4, 3)

Step 4: Intersection: (3, 2)

Step 5: Check in original equations

• 2(3) + 2 = 8 ✓
• 3 - 2 = 1 ✓

Solution: (3, 2)

Example 3: Parallel Lines (No Solution)

Solve by graphing: y = 3x + 2 y = 3x - 4

Analysis:

• Both have slope = 3 (same slope)
• Different y-intercepts (2 and -4)
• Lines are PARALLEL

Graphing: Graph y = 3x + 2 (through (0, 2) with slope 3) Graph y = 3x - 4 (through (0, -4) with slope 3)

Result: Lines never intersect

Solution: No solution (system is inconsistent)

Example 4: Identical Lines (Infinite Solutions)

Solve by graphing: y = 2x + 3 2y = 4x + 6

Step 1: Convert second equation 2y = 4x + 6 y = 2x + 3

Analysis: Both equations are identical!

• Same slope (2)
• Same y-intercept (3)
• Same line

Result: Every point on the line is a solution

Solution: Infinitely many solutions (system is dependent)

Can write as: {(x, y) | y = 2x + 3}

Using the Intercepts Method

An alternative graphing method uses x and y intercepts.

Example: Graph 3x + 2y = 12

Find y-intercept (set x = 0): 3(0) + 2y = 12 y = 6 Point: (0, 6)

Find x-intercept (set y = 0): 3x + 2(0) = 12 x = 4 Point: (4, 0)

Graph: Plot (0, 6) and (4, 0), draw line through them

Graphing with Tables

You can also create a table of values for each equation.

Example: Graph y = x + 2 and y = -2x + 5

Table for y = x + 2: | x | y | |---|---| | 0 | 2 | | 1 | 3 | | 2 | 4 |

Table for y = -2x + 5: | x | y | |---|---| | 0 | 5 | | 1 | 3 | | 2 | 1 |

Both pass through (1, 3) → Solution: (1, 3)

Estimating Non-Integer Solutions

Sometimes the intersection isn't at nice integer coordinates.

Example: y = 2x + 1 y = -x + 4

Graphing shows intersection near (1, 3)

To verify exactly: 2x + 1 = -x + 4 3x = 3 x = 1 y = 2(1) + 1 = 3

Exact solution: (1, 3) ✓

Example with decimals: y = x + 1 y = 2x - 0.5

From graph, intersection appears near (1.5, 2.5)

Solving algebraically: x + 1 = 2x - 0.5 1.5 = x y = 1.5 + 1 = 2.5

Exact solution: (1.5, 2.5)

Determining Solution Type Without Graphing

You can predict the solution type by comparing slopes and intercepts:

Compare y = m₁x + b₁ and y = m₂x + b₂:

If m₁ ≠ m₂: One solution (different slopes → lines intersect)

If m₁ = m₂ and b₁ ≠ b₂: No solution (parallel lines)

If m₁ = m₂ and b₁ = b₂: Infinite solutions (same line)

Example 1: y = 3x + 2 and y = -x + 5 m₁ = 3, m₂ = -1 (different) → One solution

Example 2: y = 2x + 1 and y = 2x - 3 m₁ = m₂ = 2, b₁ = 1, b₂ = -3 (same slope, different intercepts) → No solution

Example 3: y = x + 4 and 2y = 2x + 8 Second converts to: y = x + 4 Same equation → Infinite solutions

Graphing Calculator Tips

When using technology:

1. Enter equations in y = form
2. Adjust window to see intersection
3. Use intersection feature to find exact coordinates
4. Verify algebraically when possible

Typical window: x from -10 to 10, y from -10 to 10 Adjust if intersection is outside this range.

Real-World Application: Break-Even Analysis

Example: Company A: $100 setup fee,$5 per item Company B: $50 setup fee,$8 per item

When do they cost the same?

Let x = number of items, y = total cost

Company A: y = 5x + 100 Company B: y = 8x + 50

Graphing: Both lines intersect where costs are equal

Solving algebraically: 5x + 100 = 8x + 50 50 = 3x x = 16.67

At about 17 items, costs are approximately equal.

For fewer items: Company B is cheaper For more items: Company A is cheaper

Common Mistakes to Avoid

1. Poor graph accuracy Use graph paper or technology for precision

2. Wrong slope direction Positive slopes go up-right, negative go down-right

3. Misreading intersection Be careful with scale on axes

4. Not checking solution Graph might show (3, 4) but it could be (3, 5)

5. Assuming solution is integer Solutions can be decimals or fractions

6. Confusing parallel with identical Parallel: same slope, different intercepts (no solution) Identical: same slope AND intercept (infinite solutions)

Advantages and Limitations

Advantages of Graphing:

• Visual understanding
• Quick identification of no solution or infinite solutions
• Good for checking algebraic work
• Helps with real-world interpretation

Limitations of Graphing:

• Imprecise for non-integer solutions
• Time-consuming by hand
• Difficult with large numbers
• Requires good graphing skills

When to use graphing:

• When visual understanding is important
• To verify algebraic solutions
• When approximate solutions are sufficient
• When using graphing technology

When to use algebra instead:

• For exact solutions
• With non-integer solutions
• On tests without calculators
• When speed is important

Practice Strategy

1. Start with easy integer solutions
2. Practice identifying parallel/identical lines
3. Use substitution or elimination to verify
4. Graph by hand first, then with technology
5. Create tables when slope is unclear
6. Always check solutions
7. Practice estimating decimal solutions

Quick Reference

| Line Relationship | Slopes | Intercepts | Solution Type | |-------------------|--------|------------|---------------| | Intersecting | Different | Any | One solution | | Parallel | Same | Different | No solution | | Identical | Same | Same | Infinite solutions |

Graphing Checklist

Before you finish:

• ☐ Both equations in slope-intercept form
• ☐ Y-intercepts plotted correctly
• ☐ Slopes used accurately
• ☐ Lines extended with arrows
• ☐ Intersection point identified
• ☐ Coordinates clearly labeled
• ☐ Solution checked in both equations
• ☐ Answer written as ordered pair (x, y)