The point-slope form is an equation that represents a linear function by specifying a point on the line and the slope of the line. It is a useful way to write the equation of a line when you know a point it passes through and the slope of the line.
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The point-slope form of a linear equation is written as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope of the line.
The point-slope form is useful when you know the slope of a line and a point it passes through, as it allows you to write the equation of the line without needing to find the y-intercept.
To use the point-slope form, you substitute the known point $(x_1, y_1)$ and the slope $m$ into the equation $y - y_1 = m(x - x_1)$.
The point-slope form can be converted to the slope-intercept form, $y = mx + b$, by solving for $y$ in the point-slope equation.
Point-slope form is particularly helpful when graphing lines, as it allows you to plot the line by starting at the known point and using the slope to determine the direction and steepness of the line.
Review Questions
Explain how the point-slope form of a linear equation is different from the slope-intercept form.
The key difference between the point-slope form and the slope-intercept form of a linear equation is that the point-slope form specifies a known point on the line and the slope, while the slope-intercept form specifies the slope and the y-intercept. The point-slope form is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point and $m$ is the slope. The slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The point-slope form is useful when you know a point the line passes through and the slope, while the slope-intercept form is useful when you know the slope and the y-intercept.
Describe how to convert the point-slope form of a linear equation to the slope-intercept form.
To convert the point-slope form of a linear equation to the slope-intercept form, you need to solve the point-slope equation for $y$. The point-slope form is written as $y - y_1 = m(x - x_1)$. To get the slope-intercept form $y = mx + b$, you can rearrange the equation by adding $y_1$ to both sides to get $y = m(x - x_1) + y_1$. This simplifies to $y = mx + (y_1 - mx_1)$, where $m$ is the slope and $b = y_1 - mx_1$ is the y-intercept. By converting from the point-slope form to the slope-intercept form, you can more easily graph the line and identify key features like the slope and y-intercept.
Explain how the point-slope form can be used to write the equation of a line given a point and the slope.
The point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, is particularly useful when you know a point $(x_1, y_1)$ that the line passes through and the slope $m$ of the line. To write the equation of the line in this case, you simply plug the known point and slope into the point-slope form. For example, if you know the line passes through the point $(2, 3)$ and has a slope of $-1/2$, you would write the equation as $y - 3 = (-1/2)(x - 2)$. This allows you to easily determine the equation of the line without needing to find the y-intercept. The point-slope form is a powerful tool for writing linear equations when you have specific information about a point on the line and its slope.
The slope of a line is a measure of its steepness, calculated as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line.
A linear function is a function that has a constant rate of change, meaning its graph is a straight line.
Equation of a Line: The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of points on the line.