Start with data, not an empty regression
A regression finds the parameters of a model from a set of data points. Typing y₁ ~ mx₁ + b alone is not enough: x₁ and y₁ must already contain paired data. Follow this small example in the official Desmos calculator linked below.
1. Add and fill a table
Keep the two entries in each row together. Switching only one column changes the relationship. For this example, y increases by 2 whenever x increases by 1, so expect a slope of 2.
- Open the expression list. Select the + button at its top, then Table from the menu.
- Use the column headings x₁ and y₁. Enter one matching pair per row, as shown below.
| x₁ | y₁ |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
2. Fit the line
Check the meaning: m is the change in y for each unit of x; b is the predicted y-value at x = 0. Here, 2(4) + 1 = 9 agrees with the last row. Real data may not lie exactly on a line, so a fitted model is an approximation, not a guarantee.
- Select an empty expression row outside the table. Type y_1, press the right arrow (→) to leave the subscript, type ~mx_1, press → again, then type +b. The underscore makes the 1 a subscript.
- Use ~ to fit parameters to data. An equals sign describes an equation; it does not request this regression fit.
- Read the fitted values: m = 2 and b = 1. In another expression, enter y = 2x + 1 to see the line.
y₁ ~ mx₁ + bIf the expression shows an error
- Undefined x₁ or y₁: create and fill the table first. Match the subscripts to the actual headings; a second table may use x₂ and y₂.
- No useful fit: use at least two complete points with different x-values. Clear any earlier definitions of m or b so Desmos can estimate them.
- Points are off-screen: adjust the graph window or use Zoom Fit near the table. A hidden point is not necessarily a missing point.
3. Check a system with an intersection
Two equations, one shared solution
Find the intersection of y = 2x + 3 and y = 5x − 9.
Show worked solution
- Enter each equation on its own expression row using =. No table or regression is needed.
- Select the graphs and their intersection to read the coordinates. Adjust the window to include y = 11 if necessary.
- Verify algebraically: 2x + 3 = 5x − 9 gives x = 4; substituting into either equation gives y = 11.
Answer: (4, 11)
Choose the fastest reliable method
For 3x + 5 = 26, subtracting 5 and dividing by 3 is often quicker than making a graph. Use Desmos when a graph, a table, or a model helps you reason. Always check whether the question requests x, y, a sum, a slope, or a value in particular units. For official test preparation, practise in the SAT version linked below as well.