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Chapter 12: Problem 67

Find the slope and \(y\) -intercept of the graph of the equation. $$y=-\frac{3}{8} x+5$$

Short Answer

Expert verified
The slope of the line is -\frac{3}{8}, and the y-intercept is 5.

Step by step solution

01

Identify the slope

Begin by identifying the slope of the line. In the given equation \(y=-\frac{3}{8} x+5\), the slope \(m\) is the coefficient of \(x\), which is -\frac{3}{8}.
02

Identify the y-intercept

After identifying the slope, proceed to find the y-intercept. In the given equation \(y=-\frac{3}{8} x+5\), the y-intercept \(b\) is the constant term, which is 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope
The slope of a linear equation tells us how steep the line is and in which direction it points. When reading the equation of a line, the slope is typically represented by the coefficient of the variable
  • For example, in the equation \( y = -\frac{3}{8}x + 5 \), the slope \( m \) is \( -\frac{3}{8} \).
  • This means that for every 8 units you move horizontally to the right, you will move 3 units down vertically.
  • If the slope is positive, the line would slope upwards to the right. A negative slope like \(-\frac{3}{8}\) slopes downwards, indicating an inverse or negative relationship.
  • The larger the absolute value of the slope, the steeper the line.
Understanding slope helps in predicting how one variable changes with another. It is also crucial for graphing linear equations accurately.
y-intercept
The y-intercept of a linear equation is the point where the line crosses the y-axis. This occurs when the value of the variable \( x \) is zero.
  • In our equation example, \( y = -\frac{3}{8}x + 5 \), the y-intercept \( b \) is 5.
  • This means that the point (0, 5) is where the line will touch the y-axis.
  • The y-intercept essentially indicates the starting point of the line when graphing from the y-axis.
Visualizing the y-intercept helps to understand where the line begins and aids in sketching the rest of the line accurately from this anchor point. It also indicates the value of \( y \) when all other variables are set to zero.
Graphing Linear Equations
Graphing a linear equation involves drawing a straight line that represents all solutions of the equation on a coordinate plane.
  • Start by plotting the y-intercept. This is your starting point on the graph.
  • Use the slope to identify another point. From the y-intercept, use the rise and run method indicated by the slope. For \(-\frac{3}{8}\), you would move 3 units down and 8 units right to find the next point.
  • Draw a straight line through these points extending in both directions.
  • Label the axis and mark any intercepts to increase clarity.
This method creates a visual representation of the equation. Graphing helps in understanding the relationship between the variables and aids in predicting or finding values not easily noticeable from the equation form.

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Most popular questions from this chapter

Find the slope and \(y\) -intercept of the graph of the equation. $$2 x+3 y=8$$

A custom-illustrated sign or banner can be commissioned for a cost of 25 dollars for the material and 10.50 dollars per square foot for the artwork. The equation that represents this cost is given by \(y=10.50 x+25,\) where \(y\) is the cost and \(x\) is the number of square feet in the sign. Graph this equation for values of \(x\) from 0 to \(20 .\) The point \((15,182.5)\) is on the graph. Write a sentence that describes the meaning of this ordered pair. (GRAPH CANNOT COPY)

Do the graphs of all straight lines have a \(y\) -intercept? If not, give an example of one that does not.

Determine whether there is a line that contains all of the given points. If so, find the equation of the line. $$(-2,-4),(0,-3),(4,-1)$$

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$2 x+y=3$$
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