Write the equation of the line that passes through the origin $(0,0)$ and has a slope of $-\frac{3}{4}$ in slope-intercept form.
Answer (1)
The slope-intercept form of a line is y = m x + b , where m is the slope and b is the y-intercept.
Since the line passes through the origin ( 0 , 0 ) , the y-intercept b is 0 .
The slope of the line is given as m = − 4 3 .
Substituting these values into the slope-intercept form, we get the equation y = − 4 3 x .
y = − 4 3 x
Explanation
Understanding the Problem We are given that a line passes through the origin ( 0 , 0 ) and has a slope of − 4 3 . We need to find the equation of this line in slope-intercept form.
Recalling Slope-Intercept Form The slope-intercept form of a line is given by the equation y = m x + b , where m is the slope and b is the y-intercept.
Determining the y-intercept Since the line passes through the origin ( 0 , 0 ) , the y-intercept b is 0 . This is because the y-intercept is the point where the line crosses the y-axis, and the origin is the point ( 0 , 0 ) .
Using the Given Slope We are given that the slope m of the line is − 4 3 .
Writing the Equation Now, we substitute the values of m and b into the slope-intercept form y = m x + b . We have m = − 4 3 and b = 0 , so the equation of the line is y = − 4 3 x + 0 , which simplifies to y = − 4 3 x .
Final Answer Therefore, the equation of the line that passes through the origin ( 0 , 0 ) and has a slope of − 4 3 in slope-intercept form is y = − 4 3 x .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you're tracking the depreciation of an asset over time, a linear equation can model the decrease in value. Imagine a car that loses a fixed amount of its value each year. If the car's initial value is $20,000 and it depreciates by $2,000 each year, the car's value y after x years can be modeled by the equation y = − 2000 x + 20000 . This equation helps predict the car's value at any point in time, aiding in financial planning and decision-making.
