Give the slope and the $y$-intercept of the line with the given equation.
$y=3 x+7$
What is the slope? Select the correct choice below and fill in any answer boxes within your choice.
A. The slope is 3.
B. The slope is undefined.
What is the y-intercept? Select the correct choice below and fill in any answer boxes within your choice.
A. The y -intercept is 7.
B. There is no $y$-intercept.
Graph the equation.
Answer (2)
The equation is in slope-intercept form: y = m x + b .
Identify the slope: m = 3 .
Identify the y-intercept: b = 7 .
Graph the line using the y-intercept and slope: y = 3 x + 7 .
Explanation
Identifying the Form of the Equation The given equation is in slope-intercept form, which is y = m x + b , where m represents the slope and b represents the y-intercept.
Comparing with Slope-Intercept Form Comparing the given equation y = 3 x + 7 with the slope-intercept form y = m x + b , we can identify the slope and y-intercept.
Determining the Slope The coefficient of x in the given equation is 3. Therefore, the slope of the line is 3.
Determining the y-intercept The constant term in the given equation is 7. Therefore, the y-intercept of the line is 7. This means the line intersects the y-axis at the point (0, 7).
Graphing the Line To graph the line, we can use the slope and y-intercept. First, plot the y-intercept at (0, 7). Then, use the slope to find another point on the line. Since the slope is 3, which can be written as 1 3 , we can move 1 unit to the right and 3 units up from the y-intercept (0, 7) to find the point (1, 10).
Drawing the Line Draw a straight line through the points (0, 7) and (1, 10) to complete the graph of the equation y = 3 x + 7 .
Examples
Understanding the slope and y-intercept of a line is crucial in many real-world applications. For example, if you are tracking the cost of a taxi ride, the y-intercept might represent the initial fee, and the slope could represent the cost per mile. Similarly, in physics, the equation of motion for an object moving at a constant velocity can be represented as a linear equation, where the slope is the velocity and the y-intercept is the initial position. Linear equations are also used in economics to model supply and demand curves, where the slope and intercepts provide valuable insights into market behavior. In construction, calculating the slope of a roof or ramp is essential for ensuring proper drainage and accessibility. These are just a few examples of how understanding linear equations can help you solve practical problems in various fields.
The slope of the line given by the equation y = 3 x + 7 is 3, and the y-intercept is 7. Therefore, the correct options are A for both the slope and the y-intercept. Graphing the line involves plotting (0, 7) and (1, 10) and drawing a line through these points.
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