A line has a slope of 1/2 and passes through the point (6,7). Write its equation in slope-intercept form. Write your answer using integers, proper fractions, and improper fractions... A line has a slope of 1/2 and passes through the point (6,7). Write its equation in slope-intercept form. Write your answer using integers, proper fractions, and improper fractions in simplest form.
Understand the Problem
The question is asking to find the equation of a line in slope-intercept form (y = mx + b) given its slope and a point it passes through. We will use the slope and the point to find the y-intercept and then write the equation.
Answer
The line equation is $y = mx + (y_0 - mx_0)$.
Answer for screen readers
The equation of the line in slope-intercept form is $y = mx + b$, where $b = y_0 - mx_0$.
Steps to Solve
- Identify the slope and point
Let's assume the slope is $m$ and the point the line passes through is $(x_0, y_0)$. You'll need these values to find the y-intercept.
- Use the slope-point form of the equation
We can start with the slope-point form of a line, which is given by:
$$ y - y_0 = m(x - x_0) $$
- Rearrange to slope-intercept form
Next, we rearrange the equation to solve for $y$.
Start by expanding the equation:
$$ y - y_0 = mx - mx_0 $$
Then, add $y_0$ to both sides:
$$ y = mx - mx_0 + y_0 $$
- Simplify and find the y-intercept
Combine like terms to get the equation in slope-intercept form:
$$ y = mx + (y_0 - mx_0) $$
Here, $b = y_0 - mx_0$ represents the y-intercept.
- Write the final equation
Now, we can write the final equation of the line in the slope-intercept form:
$$ y = mx + b $$
The equation of the line in slope-intercept form is $y = mx + b$, where $b = y_0 - mx_0$.
More Information
In slope-intercept form, $y = mx + b$, $m$ represents the slope of the line, and $b$ represents the y-intercept, which is the point where the line crosses the y-axis. This format makes it easy to understand the slope and intercept visually on a graph.
Tips
- Mixing up the coordinates: Make sure to correctly identify $(x_0, y_0)$ as the coordinates of the given point.
- Forgetting to rearrange to $y = mx + b$: Always ensure the final equation is in this form to clearly see the slope and y-intercept.
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