Predicting a house's price from its size alone throws away obviously relevant information: two houses of the same size can sell for very different prices if one has more bedrooms, is much older, or sits in a different neighbourhood. Real predictions almost always draw on many features at once.

The hypothesis function you already know, h(x) = wx + b, is really just the special case of a single feature. Give every feature its own weight and add them all up, and the very same idea — a straight-line rule through the data — becomes a straight hyperplane through many features simultaneously:

Set n=1 and this collapses right back to h(x) = w_1 x_1 + b — the one-feature model you started with. Nothing about the underlying idea changed; there are simply more terms in the sum.

One clean line: the dot product

That sum of weight-times-feature is exactly a dot product of the weight vector \vec{w} = (w_1, \dots, w_n) with the feature vector \vec{x} = (x_1, \dots, x_n). So no matter how many features a model has — three or three hundred — the whole prediction collapses to one clean expression:

Here is linear algebra paying off directly: "weight each feature and add them up" is literally what a dot product computes, so the model that looks intimidating with a hundred features is exactly as simple to write down as the one with a single feature.

Worked example: predicting one house's price

Suppose training has already found the weights w_1 = 150 (per square foot), w_2 = 8{,}000 (per bedroom), w_3 = -900 (per year of age), and bias b = 20{,}000. A specific house has x_1 = 1{,}800 sq ft, x_2 = 3 bedrooms, and x_3 = 15 years old. Plug straight into the formula:

The model predicts \$300{,}500 for this house — one number, produced by weighing up size, bedrooms, and age all at once.

Reading the weights themselves

Once training is done, the weights aren't just numbers to plug in — each one tells a small story about that feature, holding everything else fixed:

• w_1 = 150 > 0 on size: a positive weight means bigger houses are predicted to cost more — every extra square foot adds $150 to the prediction.
• w_3 = -900 < 0 on age: a negative weight means older houses are predicted to cost less — every extra year of age subtracts $900, all else being equal.
• A weight close to 0 would say that feature barely moves the prediction either way, regardless of its value.

Play with the dials below: the feature values for one fixed house stay put while you adjust the weights, so you can see exactly how each weight's sign and size push the prediction up or down.

Worked example: does the extra feature actually help?

Compare two models trained on the same houses. A single-feature model uses only size: h(x) = 140x + 30{,}000. A house that is 1{,}200 sq ft but unusually new and in a great school catchment area might sell for \$225{,}000 — well above the single-feature model's guess of 140(1200) + 30000 = \$198{,}000. A multi-feature model that also sees "age" and "school rating" can pick up exactly that extra pattern and push its prediction much closer to the true price, because it has been given the information that explains the gap. Adding a genuinely predictive feature doesn't change the algorithm — it just gives it more of the picture to work with.

From a line to a hyperplane

Picture it geometrically and the jump from one feature to many is just adding dimensions. With a single feature, h(x) = w_1 x_1 + b traces a straight line through a 2-D scatter plot (feature on one axis, price on the other). Add a second feature and h(x) = w_1 x_1 + w_2 x_2 + b traces a flat plane tilted through 3-D space — one tilt for size, a different tilt for bedrooms, both baked into a single flat sheet of predictions. Keep adding features and the surface becomes a hyperplane: the same idea, just living in more dimensions than we can draw. You can't sketch a 10-dimensional hyperplane, but the algebra — one weight per axis, summed with the dot product — works exactly the same whether there are 2 features or 200.

See the plane: two features, one tilted sheet

With two input features, linear regression fits a plane through the data cloud — not a line. Each point below is a training example: its two horizontal coordinates are the features x_1 and x_2, and its height is the target y. The tilted sheet slicing through them is the model's prediction h(\vec{x}) = 0.7\,x_1 + 0.5\,x_2 — one tilt for each feature, baked into a single flat surface. Drag to rotate and watch the plane cut right through the middle of the scatter.

Same algorithm, more knobs

Nothing else changes. The cost is still mean squared error, and gradient descent still rolls downhill — just in more dimensions, adjusting every weight at once. Writing the model as \vec{w}\cdot\vec{x}+b is also why a whole dataset can be processed as one big matrix–vector multiply — fast, and exactly what hardware loves. It's also why the normal equation generalizes so cleanly from one feature to many: the same matrix algebra that solves for a single slope and intercept solves for an entire vector of weights at once.

Bias: the one term every feature shares

Notice the bias b doesn't get multiplied by any feature — it's added on its own, the same amount regardless of size, bedrooms, or age. It represents the model's baseline prediction when every feature happens to be zero (or, more usefully, it's simply the extra shift that makes all the weighted terms line up with the actual prices in the training data). With one feature this was easy to picture as "where the line crosses the price axis"; with many features it plays exactly the same role, just for a hyperplane instead of a line.

One more useful way to think of it: append a constant feature x_0 = 1 to every example, and set its weight to w_0 = b. Then the whole hypothesis — bias included — becomes a single dot product, h(\vec{x}) = \vec{w}\cdot\vec{x}, with no separate +b term left over. It's a small notational trick, but it's exactly how the maths is usually written once matrices enter the picture.