Introduction to intercepts | Algebra I | Khan Academy

Summary

The video explains how to graph linear equations, using the example y = 1/2x - 3. It starts by plotting points based on x values and calculating corresponding y values, then connects the points to form a line. The concepts of x-intercept and y-intercept are introduced, showing where the graph crosses the axes. Another equation, 5x + 6y = 30, is used to demonstrate finding intercepts by setting x or y to 0. The video emphasizes that x-intercepts occur when y is 0, and y-intercepts occur when x is 0.

Takeaways

• 📝 The video explains how to graph the linear equation y = 1/2x - 3 by plotting points and drawing a line.
• 📊 The process begins by choosing x-values, calculating corresponding y-values, and plotting points.
• 🎯 The video emphasizes that only two points are necessary to plot a line.
• 📍 The points (0, -3), (2, -2), and (4, -1) are calculated and plotted for the equation y = 1/2x - 3.
• 🔗 A line connecting these points represents the graph of the equation y = 1/2x - 3.
• 📌 The x-intercept occurs where the line crosses the x-axis, found at (6, 0).
• 📉 The y-intercept occurs where the line crosses the y-axis, found at (0, -3).
• 🧮 To find intercepts for other equations, substitute x = 0 for the y-intercept and y = 0 for the x-intercept.
• 🔄 The video demonstrates this process with another equation, 5x + 6y = 30, showing x-intercept (6, 0) and y-intercept (0, 5).
• 💡 Intercepts help identify key points on a graph, showing where the line intersects the axes.

Q & A

What is the linear equation provided in the video?

The linear equation is y = 1/2x - 3.

How do you find the y-value for a given x-value using this equation?

You substitute the x-value into the equation, calculate 1/2 of that x-value, and subtract 3 to find the corresponding y-value.

What is the y-value when x = 0?

When x = 0, the y-value is -3.

How many points are necessary to plot a straight line on a graph?

You need at least two points to plot a straight line on a graph.

What are the points plotted on the graph for the equation y = 1/2x - 3?

The points plotted are (0, -3), (2, -2), and (4, -1).

What is the x-intercept and how is it determined?

The x-intercept is the point where the graph intersects the x-axis. It occurs when y = 0. In this case, the x-intercept is at the point (6, 0).

What is the y-intercept and how is it determined?

The y-intercept is the point where the graph intersects the y-axis. It occurs when x = 0. In this case, the y-intercept is at the point (0, -3).

What does the x-intercept represent on the graph?

The x-intercept represents the point where the line crosses the x-axis, meaning the y-value is zero.

What is the general strategy for finding x and y-intercepts for any linear equation?

To find the y-intercept, set x = 0 and solve for y. To find the x-intercept, set y = 0 and solve for x.

What are the x and y-intercepts for the equation 5x + 6y = 30?

The x-intercept is (6, 0) and the y-intercept is (0, 5).

Outlines

00:00

📐 Graphing a Linear Equation

The paragraph explains how to graph a linear equation, using the example of y = 1/2x - 3. It starts by suggesting plotting points that satisfy the equation. The narrator picks easy-to-calculate x values (0, 2, and 4) and calculates the corresponding y values. The points (0, -3), (2, -2), and (4, -1) are plotted. The narrator then draws a line through these points to represent the graph of the equation. The discussion then shifts to finding the x and y-intercepts of the graph. The x-intercept is found by setting y to 0 and solving for x, which is at (6, 0). The y-intercept is the point where the graph intersects the y-axis, which is at (0, -3). The narrator encourages viewers to practice finding intercepts for other linear equations.

05:02

📉 Finding Intercepts of a Linear Equation

This paragraph continues the discussion on linear equations, focusing on finding x and y-intercepts. The equation 5x + 6y = 30 is used as an example. The narrator demonstrates how to find the y-intercept by setting x to 0 and solving for y, which results in y = 5, or the point (0, 5). To find the x-intercept, y is set to 0, leading to the equation 5x = 30, and solving for x gives x = 6, or the point (6, 0). The narrator then describes how to plot these points and draw the line that represents all the points satisfying the equation. The intercepts are reiterated as the points where the line crosses the axes, with the y-intercept at (0, 5) when x = 0, and the x-intercept at (6, 0) when y = 0.

Keywords

💡Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The general form of a linear equation is ax + by = c, where a, b, and c are constants. In the video, the linear equation y = 1/2x - 3 is used to demonstrate how to plot a line on a graph, showing that for every value of x, there is a corresponding value of y that satisfies the equation.

💡Graph

A graph is a visual representation of data, typically with points, lines, and curves. In the context of the video, the graph represents the set of all points (coordinates) that satisfy the given linear equation. The video illustrates how to plot points on a graph and then connect them to form a line, which visually represents the linear equation.

💡Coordinates

Coordinates are pairs of numbers that specify a point on a plane, typically represented as (x, y). In the video, coordinates are used to plot points on the graph that satisfy the linear equation. For example, when x = 0, the corresponding y value is calculated to be -3, resulting in the coordinate (0, -3).

💡X-intercept

The x-intercept is the point at which a line on a graph intersects the x-axis. By definition, at the x-intercept, the value of y is zero. In the video, the x-intercept of the line y = 1/2x - 3 is found to be at x = 6, y = 0. This is calculated by setting y = 0 in the equation and solving for x.

💡Y-intercept

The y-intercept is the point where a line crosses the y-axis. At the y-intercept, the value of x is zero. In the video, the y-intercept for the equation y = 1/2x - 3 is shown to be at x = 0, y = -3. This is found by substituting x = 0 into the equation.

💡Plotting

Plotting in the context of the video refers to the process of marking points on a graph that correspond to the values of a linear equation. The video demonstrates plotting by choosing specific values for x, calculating the corresponding y values, and marking these points on the graph.

💡Slope

The slope of a line is a measure of its steepness, defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. In the video, the slope of the line y = 1/2x - 3 is 1/2, indicating that for every unit increase in x, y increases by half a unit.

💡Y-value

The y-value is the vertical coordinate of a point on a graph. In the video, y-values are calculated for given x-values to determine the points that lie on the line represented by the linear equation. For instance, when x = 2, the y-value is calculated to be -2.

💡X-value

The x-value is the horizontal coordinate of a point on a graph. The video script uses x-values to find corresponding y-values that satisfy the linear equation. These pairs of x-values and y-values are then used to plot points on the graph.

💡Axis

In the context of the video, the axes refer to the x-axis and y-axis of a Cartesian coordinate system. These are the baselines from which coordinates are measured. The video discusses how to find where a line intersects these axes, which are known as the x-intercept and y-intercept.

💡Intercepts

Intercepts are the points where a line crosses the axes of a graph. The video explains how to find both the x-intercept (where the line crosses the x-axis) and the y-intercept (where the line crosses the y-axis) for a given linear equation.

Highlights

Introduction to the equation y = 1/2x - 3 and its graphical representation.

Constructing a table of values to find points on the line by choosing specific x values.

Identifying that when x = 0, y is -3, providing the first point (0, -3).

Calculating when x = 2, y becomes -2, providing the second point (2, -2).

Finding when x = 4, y becomes -1, providing the third point (4, -1).

Explanation of needing just two points to draw a line and how connecting these points forms the graph of y = 1/2x - 3.

Discussion of the x-intercept, the point where the line intersects the x-axis, calculated at (6, 0).

Introduction of the concept of intercepts, where a graph intersects the x-axis and y-axis.

Explanation of how the x-intercept is always found where y = 0, and the current x-intercept occurs at (6, 0).

Definition of the y-intercept as the point where the graph intersects the y-axis, in this case at (0, -3).

Exploration of a second equation, 5x + 6y = 30, to find its intercepts.

Demonstration of calculating the y-intercept by setting x = 0, resulting in y = 5, giving the point (0, 5).

Demonstration of calculating the x-intercept by setting y = 0, resulting in x = 6, giving the point (6, 0).

Visualization of plotting the points (0, 5) and (6, 0) for the second equation and drawing the line through them.

Final summary of how the x and y-intercepts help graph linear equations and how these intercepts are key points for visualizing relationships between variables.

Transcripts

- [Voiceover] Let's say that we have the linear equation,

y = 1/2x - 3.

So if we wanted to draw the line that represents

the set of all points, all the coordinates where the x value

and the y value satisfy this equation,

we could start off by just trying to draw,

by trying to draw a few of those points,

and then connecting them with a line.

Let's set up a little table here x, y.

And we can just try a couple of x values here,

then figure out what the corresponding y values are.

I'm going to pick x values where it's going

to be fairly easy to calculate the y values.

Let's say when x is equal to zero,

then you're gonna have 1/2 x 0 - 3,

well then y is going to be -3.

When x is, let me try x = 2, because then 1/2 x 2

is just gonna be 1.

So when x = 2, you're going to have

1/2 x 2 = 1, -3 is -2.

When x is equal to, let's try 4.

So 1/2 x 4 is 2, and then -3 is -1,

and we could keep going but actually

all we need is two points for a line.

So we're ready to plot this line if we'd like.

The point 0, -3 is on this line.

0, -3 and actually let me do this in a slightly darker color

so that we can see it on this white background.

0, -3 is on the line, 2, -2 is on the line.

So 2, -2 and then we have 4, -1.

So when x is 4, y is -1,

and I could draw a line that connects all of these

so it would look something like...

If I, let's see if I could do this.

It would look something like,

it would something like, like that.

So this right over here, this is literally,

this is the graph of y = 1/2x - 3.

Now when we look at a graph like this

an interesting thing that we might want to ask ourself

is where does the graph intersect our axes?

So first we can say, well where does

in intersect our x-axis?

When you look at this, it looks like

it happens at this point right over here.

This point where a graph intersects an axes

this is called an intercept.

This one in particular is called the x-intercept.

Why is it called the x-intercept?

Because that's where the graph

is intersecting the x-axis

and the x-intercept, it looks like

this is at the .6, 0.

Now it's very interesting, the x-intercept

happens when y = 0.

Remember, you're on the x-axis when

you haven't moved up or down from that axis

which means y = 0.

So your x-intercept happens at x = 6, y = 0.

It's this coordinate.

Now what about the y-intercept?

Well the y-intercept is this point right over here.

This is where you intersect or I guess

you could say intercept the y-axis.

So this right over here, that over there

is the y-intercept.

The y-intercept is at the coordinate

that has a 0 for the x-coordinate.

X is 0 here and y is -3.

X is 0 and y is -3.

This was actually one of the points,

or one of the pairs that we first tried out.

You can validate that 6, 0 satisfies this equation

right over here.

If x is 6, 1/2 x 6 is 3, -3 is indeed equal to 0.

So now that we know what an x-intercept is,

it's the point where a graph intersects the x-axis

or intercepts the x-axis and the y-intercept

is the point where a graph intercepts the y-axis

or intersects the y-axis.

Let's try to see if we can find the x and y-intercepts

for a few other linear equations.

So let's say that I had the linear equation.

Let's say that I have 5x + 6y = 30.

I encourage you to pause this video,

and figure out what are the x and y-intercepts

for the graph that represents the solutions,

all the xy pairs that satisfy this equation.

Well the easiest thing to do here,

let's see what the y value is when x = 0

and what x value is when y = 0.

When x = 0 this becomes 6y = 30.

So 6 times what is 30?

Well y would be equal to 5 here.

So when x is 0, y is 5.

What about when y is 0?

Well when y is 0, that's going to be 0,

and you have 5x = 30.

Well then x would be equal to 6.

Then x would be equal to 6.

So we could plot those points, 0, 5.

When x is 0, y is 5.

When x is 6, y is 0.

So those are both points on this graph

and then the actual graph is going to,

or the actual line that represents the x and y pairs

that satisfy this equation is going to look like,

it's going to look like this.

I'll just try.

So I can make it go, it's going to look like...

It's going to go through those two points.

So it going to...I can make it go the other way too.

Let me see.

It's going to go through those two points

and so it's going to look something like that.

Now what are its' x and y-intercepts?

Well, we already kind of figured it out

but the intercepts themselves,

these are the points on the graph where

they intersect the axes.

So this right over here, this is the y-intercept.

That point is the y-intercept

and it happens, it's always going to happen

when x = 0, and when x = 0 we know that y = 5.

It's that point, the point 0, 5.

And what is the y inter...what is the x-intercept?

The x-intercept is the point, it's actually

the same x-intercept for this equation right over here.

It's the point 6, 0. That point right over there.