Why the Intersection Point Is the Solution

Lesson 1 of 3: Systems of Linear Equations

In this lesson:

• See why two lines crossing gives the answer
• Tell one, none, or infinitely many apart

Learning Objectives for This Lesson

You should be able to:

1. Explain what it means to solve a system
2. Graph two lines and find their intersection
3. Verify the intersection satisfies both equations
4. Classify a system by its number of solutions
5. Explain why graphing can be imprecise

Can One Pair Satisfy Both?

Find numbers and that make both true:

Try a few pairs. What works in one might fail the other.

Testing Pairs One at a Time

Try :

• First equation:
• Second equation:

Both give , so works.

The Pair Is Where the Lines Cross

The lines meet at — the pair you found.

A Line Holds Every Solution

• Every point on a line makes its equation true
• Every point off the line makes it false
• A line is the full picture of all solutions

Why the Crossing Solves Both

• The crossing point sits on both lines
• On line one, so it satisfies equation one
• On line two, so it satisfies equation two
• One point, both equations true

Worked Example: Graph and Verify

Solve and :

• Graph both lines, find where they meet
• They cross at
• Check: and

Does This Other Point Work?

Test in both equations:

• : — true
• : — false

It solves one, not both.

Quick Check: Find the Solution

Solve and by graphing.

Graph both, find where they cross, then verify your point in both equations.

Do Two Lines Always Cross?

Before graphing, compare slopes:

Both have slope 2. What does that tell you about whether they meet?

Case One: Different Slopes, One Solution

• Different slopes means the lines tilt differently
• Lines tilting differently must cross once
• Exactly one point lies on both
• One solution

Case Two: Parallel Lines, No Solution

Same slope, different starts — they never meet.

No Solution in the Algebra

Setting the equations equal:

Subtract : you get , which is false.

Case Three: Same Line, Infinitely Many

Infinitely Many in the Algebra

Both equations simplify to .

Setting them equal gives , or .

Always true means every works.

Slopes and Intercepts Decide the Case

Predict the Case Without Graphing First

How many solutions? Use slopes and intercepts:

1. and
2. and
3. and

What If the Crossing Isn't on a Corner?

Every crossing so far landed on a grid point. Try reading this one:

Where exactly do they cross? Could two people disagree?

A Clean Case Reads Exactly

Solve and :

• They cross at
• Check: and

Integer coordinates read cleanly off the grid.

When Estimates Disagree, Use Algebra

Set : then , so , .

Two Tools, Two Different Jobs

• Graphing shows the big picture and the case
• Algebra gives the exact answer
• Use graphing to understand, algebra to be precise

Find the Error in This Graph

A student reads the crossing as for and .

Check : true, but false.

What went wrong?

Commit to a Prediction on Your Own

Without graphing, how many solutions?

Write your answer and one reason on your own before we reveal it.

Your Turn: The Full Solution

Graph and .

Find the intersection, then verify it in both equations.

Three Common Mistakes to Avoid

Close slopes still cross — only equal slopes are parallel
Two straight lines meet at most once
"No solution" is a real answer, not a failure

Key Takeaways From This Lesson

✓ A line shows every solution to its equation
✓ The crossing solves both equations at once
✓ Three cases: one, none, infinitely many
✓ Always verify by substituting into both

Coming Up Next: The Algebra Methods

In Lesson 2, you'll learn substitution and elimination — algebra that pins down the exact answer every time, even when the crossing falls between grid lines.