Examples of slope-intercept form in the following topics:
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- One of the most common representations for a line is with the slope-intercept form.
- Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and y-intercept.
- Simply substitute the values into the slope-intercept form to obtain:
- Let's write the equation 3x+2y=−4 in slope-intercept form and identify the slope and y-intercept.
- Now that the equation is in slope-intercept form, we see that the slope m=−23, and the y-intercept b=−2.
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- Example: Write the equation of a line in point-slope form, given a point (2,1) and slope −4, and convert to slope-intercept form
- To switch this equation into slope-intercept form, solve the equation for y:
- Example: Write the equation of a line in point-slope form, given point (−3,6) and point (1,2), and convert to slope-intercept form
- Graph of the line y−1=−4(x−2), through the point (2,1) with slope of −4, as well as the slope-intercept form, y=−4x+9.
- Use point-slope form to find the equation of a line passing through two points and verify that it is equivalent to the slope-intercept form of the equation
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- Before successfully solving a system graphically, one must understand how to graph equations written in standard form, or Ax+By=C.
- To do this, you need to convert the equations to slope-intercept form, or y=mx+b, where m = slope and b = y-intercept.
- The best way to convert an equation to slope-intercept form is to first isolate the y variable and then divide the right side by B, as shown below.
- Now −BA is the slope m, and BC is the y-intercept b.
- Once you have converted the equations into slope-intercept form, you can graph the equations.
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- Recall that the slope-intercept form of an equation is: y=mx+b and the point-slope form of an equation is: y−y1=m(x−x1), both contain information about the slope, namely the constant m.
- Example: Write an equation of the line (in slope-intercept form) that is parallel to the line y=−2x+4 and passes through the point (−1,1)
- Start with the equation for slope-intercept form and then substitute the values for the slope and the point, and solve for b: y=mx+b.
- Example: Write an equation of the line (in slope-intercept form) that is perpendicular to the line y=41x−3 and passes through the point (2,4)
- Again, start with the slope-intercept form and substitute the values, except the value for the slope will be the negative reciprocal.
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- A linear equation written in standard form makes it easy to calculate the zero, or x-intercept, of the equation.
- For example, consider an equation in slope-intercept form: y=−12x+5.
- We know that the y-intercept of a linear equation can easily be found by putting the equation in slope-intercept form.
- However, the zero, or x-intercept of a linear equation can easily be found by putting it into standard form.
- Note that the y-intercept and slope can also be calculated using the coefficients and constant of the standard form equation.
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- Linear functions are algebraic equations whose graphs are straight lines with unique values for their slope and y-intercepts.
- For example, a common equation, y=mx+b, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with x and y as variables and m and b as constants.
- Vertical lines have an undefined slope, and cannot be represented in the form y=mx+b, but instead as an equation of the form x=c for a constant c, because the vertical line intersects a value on the x-axis, c.
- Horizontal lines have a slope of zero and is represented by the form, y=b, where b is the y-intercept.
- The blue line has a positive slope of 21 and a y-intercept of −3; the red line has a negative slope of −1 and a y-intercept of 5.
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- Slope is often denoted by the letter m.
- Recall the slop-intercept form of a line, y=mx+b.
- Putting the equation of a line into this form gives you the slope (m) of a line, and its y-intercept (b).
- In other words, a line with a slope of −9 is steeper than a line with a slope of 7.
- We can see the slope is decreasing, so be sure to look for a negative slope.
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- A common form of a linear equation in the two variables x and y is:
- Where m (slope) and b (intercept) designate constants.
- The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane.
- In this particular equation, the constant m determines the slope or gradient of that line, and the constant term b determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.
- Three lines — the red and blue lines have the same slope, while the red and green ones have same y-intercept.
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- In the regression line equation the constant m is the slope of the line and b is the y-intercept.
- Regression analysis is the process of building a model of the relationship between variables in the form of mathematical equations.
- The constant m is slope of the line and b is the y-intercept -- the value where the line cross the y axis.
- An equation where y is the dependent variable, x is the independent variable, m is the slope, and b is the intercept.
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- The concepts of slope and intercept are essential to understand in the context of graphing data.
- The slope or gradient of a line describes its steepness, incline, or grade.
- A higher slope value indicates a steeper incline.
- It also acts as a reference point for slopes and some graphs.
- Functions which are undefined at x=0 have no y-intercept.