Predict, then look
A small sketch can hide a crossing.
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Step 1 of 4
- A sketch square, 0 to 10 both ways.
- Line 1:
2x + 3y = 28. - Line 2:
4x + 7y = 58. They lean almost alike here. - The exact meet is (11, 2), one square outside.
The rule
A sketch shows a small part of the plane. So it can miss a far crossing. Compare the ratios first. Use the picture to check. Use the algebra for the exact answer.
Example
The pair is 2x + 3y = 28 and 4x + 7y = 58. The ratios 2/4 and 3/7 are not equal. So the lines cross once.
To find where, double the first equation: 4x + 6y = 56. Subtract it from 4x + 7y = 58. That leaves y = 2, since 7y − 6y = y and 58 − 56 = 2. Use that in the first equation: 2x + 6 = 28. So 2x = 22 and x = 11.
Check both. 2×11 + 3×2 = 22 + 6 = 28. 4×11 + 7×2 = 44 + 14 = 58.
| Tool | What it gives you | What it cannot do |
|---|---|---|
| The ratio test | The verdict, before work | Give the actual pair |
| A drawn picture | A rough crossing | Show a far crossing |
| The algebra | The exact pair | Nothing. It is exact. |