The x-intercept of a linear equation

Example 1. Omar is building a tree house. For every plank that he puts on, Omar is using four nails. Define a linear equation to represent the relation between number of planks and number of nails.

We can create a graph or an equation to represent the linear equation. For this, we need to find the slope and the y-intercept.

Part 1: Set up.

We can describe the situation using a linear equation on a graph where the y-axis represents the number of nails and the x-axis represents the number of planks.

Part 2: Find the y-intercept.

The y-intercept is the value of y that corresponds to when x is 0.

If x is 0, then there are no planks. How many nails would Omar need to put up 0 planks? The answer is none! Omar needs 0 nails to put up 0 planks.

So, in this case, the y-value that corresponds to when x is 0 is also 0. That means that the linear equation crosses the y-axis at 0. We can plot the y-intercept in the graph.

Part 3: Find the slope.

As we move to the right of the graph, we are increasing the number of planks. We are told that for every plank, Omar is using 4 nails. Therefore, every time we move 1 unit to the right, the line climbs up 4 units.

The slope is defined as how much we moved vertically over how much we moved horizontally.

So the slope of the linear equation is 4.

Part 4: The linear equation.

In conclusion, we have the following linear graph to describe the relation of the number of planks that Omar is using for this tree house and the number of nails he uses.

The equation will look like this, where y represents the number of nails and the x represents the number of planks.

Since the y-intercept is 0 and the slope is 4, the equation is:

The equation can be simplified as:

Example 2. Let's say that Jane Lane repeats her interstellar trip with a group of friends. Suppose that Jane brought 8 packs of frozen food again, but the group goes through 4 packs every 2 weeks.

Define a linear equation to represent the relation between the number of food packs left and the number of weeks on the trip. In this case, what does the x-intercept represent?

To create a graph or an equation that represents the linear equation, we need to find the slope and the y-intercept.

Part 1: Set up.

We can represent the relation between frozen food packs and weeks on the trip as a graph whose axes represent the number of food packs and the number of weeks spent on the trip.

Part 2: The y-intercept.

We saw in the last exercise that the y-intercept is the number of food packs when the expedition has spent 0 weeks in space. In other words, the number of food packs that they started out with, which is 8.

Therefore, the y-intercept is 8 and the line will cross the y-axis at 8.

Part 3: The slope.

How does the amount on the y-axis (food packs) changes as the amount on the x-axis (weeks) changes? The slope tells us how quickly or slowly Jane and her friends use the frozen food packs.

In this case, we are told that every two weeks, Jane and her friends use 4 food packs. So if we move two units to the right, the line drops down by 4 units.

The slope is defined as how much we moved vertically over how much we moved horizontally.

So the slope of the linear equation is -2.

Part 4: Conclusion.

The equation will look like this, where y represents the number of food packs left and the x represents the number of weeks on the galactic trip.

Since the y-intercept is 8 and the slope is -2, the equation is:

And it can be represented as the following graph:

Part 5: Interpretation of the x-intercept.

Since the line crosses the x-axis at x = 4, that means that the after 4 weeks, Jane and her friends have used up all their frozen food packs. So the x-intercept in this case represents the number of weeks that the frozen food packs will last them.