Another Look at Linear Equations: Algebra Chapter 2 Section 4
Hello internet, and welcome to the Algebra Lecture Series from The Science of Life. This entry is an introduction to Linear Functions in both graphs and models.
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There are two situations where the slope would not be stated in the slope-intercept form. These are cases where the line crosses only one of the 2 (or more) axes.
First, let's consider the horizontal line. If we have a line which has a y-intercept but does not have an x-intercept (does not cross the x-axis), it is called a horizontal line. Let's look at it from the point of view of the slope-intercept form we developed last time, $y=mx+b$. We know that the slope is the change in the y-value per change in the x-value. If we have two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$, the slope is mathematically defined as $m=\frac{y2-y1}{x2-x1}$. The denominator, the $ dx=x2-x1$, is never going to be zero when there's a y-intercept, since we can go from the left of the y-intercept to the right of the y-intercept through the intercept, exclusively by changing the value of the x-variable. The y-variable, however, will never change. If the line is always horizontal, then there is no way to get a single x-intercept; either we'll have no x-intercept (the line is parallel to the x-axis) or an infinite amount of x-intercepts (the line is on the x-axis). This means that $y_{2}=y_{1}$, or $y_{2}-y_{1}=0$. so pluging this into the slope, we get that $m=\frac{y_{2}-y_{1}}{x2-x1}=\frac{0}{x2-x1}=0$. Plugging this slope into the slope intercept form will give us $y=mx+b=0*x+b$. Since multiplying anything finite by zero will equal zero, this gives us $y=0+b$. Since anything added to zero is that anything, this gives us $y=b$, or the y-value will always be the value of the intercept.
The vertical line is one which crosses the x-axis (has an x-intercept) without crossing the y-axis. It is described by the equation x=a. Since any two points will have the same x-value and only changes in the y-variable, it's slope is given by $m=\frac{y_{2}-y_{1}}{0}$. Since a division by 0 is nonsensical, this means that a vertical has an undefined slope. So all vertical lines have an undefined slope.
Looking at lines that have defined slopes other than 0, we can determine the intercept of one variable by setting the other variable equal to zero. If we want to find the y-intercept, we would take the equation $y=mx+b$ and set $x=0$ and visa versa. This makes sense since b is defined as the y-intercept. So the intercept for y is b and the intercept for x is $x=\frac{-b}{m}$.
We've seen the slope-intercept form of the linear function; there is a second commonly used equation of the linear function which is called the Standard Form of a Line. This is in the form of $Ax+By=C$, where A, B, and C are all real numbers and A and B are both non-zero. This is commonly used when we deal with situation where the sum of two variables cannot exceed a certain value C.
First, let's consider the horizontal line. If we have a line which has a y-intercept but does not have an x-intercept (does not cross the x-axis), it is called a horizontal line. Let's look at it from the point of view of the slope-intercept form we developed last time, $y=mx+b$. We know that the slope is the change in the y-value per change in the x-value. If we have two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$, the slope is mathematically defined as $m=\frac{y2-y1}{x2-x1}$. The denominator, the $ dx=x2-x1$, is never going to be zero when there's a y-intercept, since we can go from the left of the y-intercept to the right of the y-intercept through the intercept, exclusively by changing the value of the x-variable. The y-variable, however, will never change. If the line is always horizontal, then there is no way to get a single x-intercept; either we'll have no x-intercept (the line is parallel to the x-axis) or an infinite amount of x-intercepts (the line is on the x-axis). This means that $y_{2}=y_{1}$, or $y_{2}-y_{1}=0$. so pluging this into the slope, we get that $m=\frac{y_{2}-y_{1}}{x2-x1}=\frac{0}{x2-x1}=0$. Plugging this slope into the slope intercept form will give us $y=mx+b=0*x+b$. Since multiplying anything finite by zero will equal zero, this gives us $y=0+b$. Since anything added to zero is that anything, this gives us $y=b$, or the y-value will always be the value of the intercept.
The vertical line is one which crosses the x-axis (has an x-intercept) without crossing the y-axis. It is described by the equation x=a. Since any two points will have the same x-value and only changes in the y-variable, it's slope is given by $m=\frac{y_{2}-y_{1}}{0}$. Since a division by 0 is nonsensical, this means that a vertical has an undefined slope. So all vertical lines have an undefined slope.
Looking at lines that have defined slopes other than 0, we can determine the intercept of one variable by setting the other variable equal to zero. If we want to find the y-intercept, we would take the equation $y=mx+b$ and set $x=0$ and visa versa. This makes sense since b is defined as the y-intercept. So the intercept for y is b and the intercept for x is $x=\frac{-b}{m}$.
We've seen the slope-intercept form of the linear function; there is a second commonly used equation of the linear function which is called the Standard Form of a Line. This is in the form of $Ax+By=C$, where A, B, and C are all real numbers and A and B are both non-zero. This is commonly used when we deal with situation where the sum of two variables cannot exceed a certain value C.
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