How to find the slope and y intercept of an equation

To find the y intercept using the equation of the line, plug in 0 for the x variable and solve for y. If the equation is written in the slope-intercept form, plug in the slope and the x and y coordinates for a point on the line to solve for y. If you don't know the slope, calculate it by dividing the rise of the line by the run. If you want to find the y-intercept if you only know 2 points along the line, keep reading the article!

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In this lesson, we learn how to graph our line using the y-intercept and the slope. First, we know that the y-intercept (b) is on the y-axis, so we graph that point. Next, we use the slope to find a second point in relation to that intercept. The following video will show you how this is done with two examples.

Video Source (05:37 mins) | Transcript

Steps for graphing an equation using the slope and y-intercept:

1. Find the y-intercept = b of the equation y = mx + b.
2. Plot the y-intercept. The point will be (0, b).
3. Find the slope=m of the equation y = mx + b.
4. Make a single step, using the rise and run from the slope. (Make sure you go up to the right if it’s positive and down to the right if it’s negative.)
5. Connect those two points with your line.

Additional Resources

• Khan Academy: Intro to Slope-intercept Form (08:59 mins, Transcript)
• Khan Academy: Graph from Slope-intercept Equations (03:01 mins, Transcript)
• Khan Academy: Slope-intercept Examples (03:45 mins, Transcript)

Practice Problems

1. Plot the line \({\text{y}}=-3{\text{x}}+2\) starting with the y-intercept and then using the slope.
2. Plot the line \({\text{y}}=\frac{1}{2}{\text{x}}-3\) starting with the y-intercept and then using the slope.
3. Plot the line \({\text{y}}=-\frac{3}{5}{\text{x}}+1\) starting with the y-intercept and then using the slope.
4. Plot the line \({\text{y}}=2{\text{x}}+3\) starting with the y-intercept and then using the slope.
5. Plot the line \({\text{y}}=-{\text{x}}-4\) starting with the y-intercept and then using the slope.
6. Plot the line \({\text{y}}=\frac{4}{5}{\text{x}}+4\) starting with the y-intercept and then using the slope.

Solutions

1. Note: The graph you create may look slightly different depending on the spacing you choose for your x and y-axis. The correct graph should still have the same direction of slope and the x and y-intercepts should be the same.
   
   
   (Written Solution)

   To graph this line we need to identify the slope and the y-intercept. The equation is written in slope-intercept form, y=mx+b, where m is the slope and b is the y-intercept.

   Step 1: Find the slope and the y-intercept of the line:

   The equation of the line is

   \({\text{y}}={\color{red}-3}{\text{x}}{\color{Blue}+2}\)

   So the slope is \({\color{red}-3}\), and the y-intercept is 2.

   Step 2: Graph the y-intercept:

   

   This is a picture of a coordinate plane with the points \((0, -4)\) and \((1, -5)\) graphed on it. There is a line passing through both points.

   If you're seeing this message, it means we're having trouble loading external resources on our website.

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   The slope and y-intercept calculator takes a linear equation and allows you to calculate the slope and y-intercept for the equation. The equation can be in any form as long as its linear and and you can find the slope and y-intercept.

   If you're seeing this message, it means we're having trouble loading external resources on our website.

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   Hello! Today we are going to take a look at how to find the slope-intercept form of a linear equation when given a point on the line and the slope of the line, and when we are given two points on a line. Remember, the slope-intercept form of a linear equation is:

   \(y=mx+b\)

    

   where \(m\) stands for our slope, and \(b\) stands for the \(y\)-intercept.

   Let’s start with finding the equation when given a point on the line and the slope of the line. So, let’s say we’re given the point \((3,5)\).

   Point: \((3,5)\)

    

   This means that the point \((3,5)\) is a point that’s on the line. And let’s say that we’re also given that our slope of the line is 2.

   Slope: \(2\)

    

   So, the first thing we’re going to do is write out the slope-intercept form equation of the line:

   \(y=mx+b\)

    

   This helps us remember the formula, and it makes sure we plug in values into the correct places. Now, to create this equation, we need to plug in values for both \(m\) and \(b\). Remember, \(m\) stands for the slope, which we are given. The slope is 2. So we’re going to plug in this value.

   \(y=2x+b\)

    

   Now we need to find \(b\). To find \(b\), we are going to plug in the point we are given for \(x\) and \(y\). The point \((3,5)\) stands for \((x,y)\), so \(x=3\) and \(y=5\). So, we’re going to plug in these values and then solve for \(b\).

   \(5=2(3)+b\)

    

   And now we solve for \(b\). Start by multiplying 2 and 3 on the right side.

   \(5=6+b\)

    

   Next, subtract 6 from both sides.

   \(-1=b\)

    

   So, now that we know that our \(y\)-intercept is \(-1\), we can go back to our equation over here and plug in this value.

   \(y=2x+(-1)\)

    

   Now remember, adding a negative number is the same thing as subtracting, so we can rewrite this as:

   \(y=2x-1\)

    

   And that’s our final answer! Let’s try another one.

   Find the slope-intercept equation of a line that passes through the point \((-6,2)\) and has a slope of 5.

   First, write out the general form of a slope-intercept equation.

   \(y=mx+b\)

    

   Then, plug in the slope. Our slope is 5, so we’ll get:

   \(y=5x+b\)

    

   Now we’re going to plug in our point for \(x\) and \(y\).

   \(2=5(-6)+b\)

    

   Now, solve for \(b\). Start by multiplying 5 and -6.

   \(2=-30+b\)

    

   And now we’ll add 30 to both sides of our equation. That gives us:

   \(32=b\)

    

   So our \(y\)-intercept is 32. Now we can go back here and plug this value in for \(b\).

   \(y=5x+32\)

    

   Now that we’ve got the hang of that, let’s take a look at how to find the slope-intercept equation when we’re given two points on the line. So, let’s say we’re given the points:

   Points: \((6,5)\) and \((7,1)\)

    

   First, write out the general form of a slope-intercept equation.

   \(y=mx+b\)

    

   Now we need to plug in values for \(m\) and \(b\). This time we aren’t given \(m\), but we can solve for it using the slope formula.

   \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

    

   Let’s use the point \((6,5)\) as \((x_{1},y_{1})\) and the point \((7,1)\) as \((x_{2}, y_{2})\). If you switched the points and said that \((6,5)\) was \((x_{2},y_{2})\) and \((7,1)\) was \((x_{1},y_{1})\), that would also be okay, you’ll get the same slope value either way. Now, let’s plug in our values and find our slope.

   \(m=\frac{1-5}{7-6}=\frac{-4}{1}=-4\)

    

   So our slope \((m)\) is \(-4\). Now that we know this, we can follow the same steps as we did in our previous practice problems. Plug in \(-4\) into the equation for \(m\).

   \(y=-4x+b\)

    

   For this next part, where we plug in a point, you can use either point. If you do it correctly, you will get the same answer both ways. For this example, let’s use the point \((6,5)\).

   \(5=-4(6)+b\)

    

   Now we’re going to start by multiplying \(-4\) and \(6\).

   \(5=-24+b\)

    

   Then, we’ll add 24 to both sides.

   \(29=b\)

    

   So, since our \(y\)-intercept, \(b\), is equal to 29, we can now go over here and plug that into our equation. So we’ll have:

   \(y=-4x+29\)

    

   And that’s our final answer! Before we go, I want to try one more problem.

   Find the point-slope equation of a line that passes through the points \((-9,11)\) and \((12,18)\).

   First things first, write out your general equation.

   \(y=mx+b\)

    

   Then, we solve for \(m\) using the slope formula.

   \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

    

   If we plug in our points, we’ll get:

   \(m=\frac{18-11}{12-(-9)}\)

    

   We can rewrite this denominator as, \(12+9\), since subtracting a negative number is the same as adding that number.

   \(m=\frac{18-11}{12+9}=\frac{7}{21}\)

    

   Then we can simplify this fraction by dividing both the numerator and denominator by 7 and get:

   \(m=\frac{1}{3}\)

    

   So, our slope is equal to \(\frac{1}{3}\). Now that we have our slope, we can plug it into our equation.

   \(y=\frac{1}{3}x+b\)

    

   Next, choose either point to plug in for \(x\) and \(y\). This time, let’s use the point \((12,18)\).

   \(18=\frac{1}{3}(12)+b\)

    

   Now we solve for \(b\). Start by multiplying 13 and 12.

   \(18=4+b\)

    

   And then we’ll subtract 4 from both sides.

   \(14=b\)

    

   So, now that we know that our \(y\)-intercept is equal to 14, we can come back over here and plug it into our equation and get:

   \(y=\frac{1}{3}x+14\)

    

   And that’s all there is to it! I hope this video helped you learn how to find the slope-intercept form of a linear equation. Thanks for watching, and happy studying!

   Practice Questions

   Question #1:

    
   Use the slope-intercept form to write the equation of a line that has a slope of –2 and contains the point \((4,-3)\).

   \(y=-2x-5\)

   \(y=-2x-7\)

   \(y=-2x+5\)

   \(y=-2x-11\)

   Show Answer

   Answer:

   The equation of a line in slope-intercept form is:

   \(y=mx+b\)

   • \(m\) is the slope of the line.
   • \(b\) is the \(y\)-intercept of the line.
   • \((x,y)\) is a point on the line.

   First, substitute \(m=-2\) into the equation.

   \(y=-2x+b\)

   Next, substitute the coordinates of the given point, \(x=4\) and \(y=-3\) into the equation and solve for \(b\).

   \(-3=-2\left(4\right)+b\)
   \(-3=-8+b\)
   \(-3+8=-8+b+8\)
   \(5=b\)

   Lastly, substitute the value of \(b\) into the equation to write the linear equation of the line in slope-intercept form.

   How do you find the y

   The slope intercept formula y = mx + b is used when you know the slope of the line to be examined and the point given is also the y intercept (0, b). In the formula, b represents the y value of the y intercept point.

   How do you find the y

   To find y-intercept: set x = 0 and solve for y. The point will be (0, y). To find x-intercept: set y = 0 and solve for x. The point will be (x, 0).

   How do you find the slope and slope of a equation?

   To find the slope of a line given the equation of the line, first write it in slope-intercept form. Use inverse operations to solve for y so that it is written as y=mx+b. Then you can easily see the slope since it is the coefficient of the x variable, or the number in front of x.