Equations you can see

Solve equations visually

Every equation is a question: where do two things become equal? Draw each side as a line, and the answer is where the lines cross. Drag the line and watch Pip ride to the solution of ax + b = 7.

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2x + 1 = 7

Slope a 2

Start b 1

2x + 1 = 7
2x = 6
x = 6 ÷ 2 = 3

The blue line meets the flat line at (3, 7), so x = 3.

Idea 1 · Where do they meet?

Where do they meet? An equation as two lines

An equation like 2x + 1 = 7 asks one question: for which x does the left side equal the right side? Draw each side as a line, and the answer is where the two lines cross.

A number on its own, like 7, is a line too: a flat one, because it gives 7 whatever x is.

Aha!

A number on its own, like 7, is also a line: a flat one, because it gives 7 whatever x is. Making 2x + 1 equal to 7 gives you the x where the two lines cross.

Idea 2 · x on both sides

x on both sides: two sloped lines

When x is on both sides, like 3x + 1 = x + 7, draw both sides as lines. Now both lines slope, but the question is the same: where do they cross?

The crossing is at x = 3, because 3 × 3 + 1 and 3 + 7 are both 10.

Aha!

Making two lines equal gives you the solution for x, which is where they cross each other.

Idea 3 · Fractions in the answer

Fractions in the answer: between the grid lines

Not every answer is a whole number. In x + 2 = 9 − 2x the lines cross between x = 2 and x = 3.

Solving gives the exact answer, x = 7/3, even though you could never read it exactly off the grid.

Aha!

Solving gives the exact crossing even when it falls between two grid lines.

Idea 4 · No solution, every solution

No solution, or every solution

Two lines with the same slope run side by side and never meet. So an equation like 2x + 1 = 2x + 4 has no solution at all.

But if the two sides are really the same line, like 2x + 1 = 2x + 1, then every x is a solution.

Aha!

Two lines with the same slope never meet, so their equation has no solution. If they are the same line, every x is a solution.

Quick answers

Questions about solving equations

How can you solve an equation with a graph?

Draw each side of the equation as a line. The x-value where the two lines cross is the solution, because that is where both sides are equal. For 2x + 1 = 7, the lines y = 2x + 1 and y = 7 cross at x = 3.

What if the two lines never cross?

Lines with the same slope but different starting heights are parallel and never meet. The equation then has no solution, for example 2x + 1 = 2x + 4.

What if both sides are the same line?

Then every x makes both sides equal, so every number is a solution, for example 2x + 1 = 2x + 1.

Can the answer to an equation be a fraction?

Yes. The lines can cross between the grid lines. For x + 2 = 9 − 2x the lines cross at x = 7/3, which is about 2.33.

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Solve it with Pip, a math game for adults

In Pip: Math Adventure, World 2 is all about equations. Instead of pushing symbols around a page, you work out where lines meet, and Pip acts out your answer. A wrong one shows you why it fails.

It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.

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