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Chapter 10: Problem 30

Draw the line described. With \(x\) intercept \(-3\) and with \(m=0.25\)

Short Answer

Expert verified
The line equation is \(y = 0.25x + 0.75\).

Step by step solution

01

Understanding Line Properties

We are given that the line has an x-intercept at \(-3\) and a slope \(m = 0.25\). The equation of a line can be generally described by the slope-intercept form \(y=mx+c\).
02

Determine the Intercept Form

Since the line intercepts the x-axis at \(-3\), the point on the line would be \((-3, 0)\). We will use this point to find the slope-intercept equation.
03

Insert Known Values into the Equation

Substitute the point and slope into the general equation \(y = mx + c\): \[0 = 0.25(-3) + c\]
04

Solve for the Y-Intercept

To find \(c\), solve the equation: \[0 = -0.75 + c\]. Simplify to find \[c = 0.75\].
05

Write the Final Equation

Now that we know the slope \(m = 0.25\) and y-intercept \(c = 0.75\), the equation of the line is: \(y = 0.25x + 0.75\).
06

Graph the Line

Using the line equation, plot points starting from the y-intercept (0, 0.75) and follow the slope rise of 0.25 over a run of 1, then connect these points to graph the line.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a widely used method for expressing equations of lines. It is written as \(y = mx + c\), where \(m\) represents the slope, and \(c\) signifies the y-intercept. This form is particularly useful because it provides immediate insight into the steepness of the line and where it crosses the y-axis.
  • Slope (\(m\)): It shows how steep the line is. If the slope is positive, the line goes upwards as it moves from left to right. If negative, it goes downwards.
  • Y-Intercept (\(c\)): This is the point where the line crosses the y-axis. It tells us what the value of \(y\) is when \(x\) equals zero.
Understanding these components helps us easily translate real-world scenarios into graphical representations using lines. The slope-intercept form simplifies this by presenting all necessary information in a clear manner, making it easier to visualize and graph the line.
X-Intercept
The x-intercept is a crucial aspect of linear equations, representing the point where the line crosses the x-axis. In mathematical terms, it is defined as the value of \(x\) when \(y\) is zero.
  • To find the x-intercept from an equation, set \(y\) to zero and solve for \(x\).
  • It provides insight into the root or zero of the linear equation, which is crucial in various applications including algebraic problem-solving and analysis.
In the given exercise, the x-intercept was specified as \(-3\). This meant that our line passes through the point \((-3, 0)\). By knowing the x-intercept, we effectively determine one anchor point for plotting the line on a graph. It's an essential piece for drawing accurate representations of linear relationships.
Graphing Lines
Graphing lines involves plotting points on a coordinate plane to represent linear relationships described by equations. Once you have your line's equation in slope-intercept form, it's straightforward to begin plotting.
  • Start at the y-intercept, which in this exercise is \(0.75\).
  • Use the slope to determine the next points. With a slope of \(0.25\), rise 0.25 units for every 1 unit you run to the right.
  • Draw the line through these plotted points.
This method provides a visual representation of the mathematical relationship, helping in both understanding and analysis. When graphed, lines reveal trends and relationships between variables, making it easier to interpret the data and solve equations. By using these steps, one can accurately graph any line described by a linear equation.

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Most popular questions from this chapter

Use substitution to solve the system. $$y=\frac{1}{2} x-3 \text { and } y=\frac{1}{3} x-2$$

Use slopes to verify that the graphs of the equations $$A x+B y=C \quad \text { and } \quad A x+B y=D$$ are parallel. (NOTE: \(A \neq 0, B \neq 0, \text { and } C \neq D .)\)

Find the equation of the line described. Leave the solution in the form \(A x+B y=C\). The line is the perpendicular bisector of the line segment that joins \((-4,5)\) and \((1,1)\).

Points \(A\) and \(B\) have symmetry with respect to either the \(x\) axis or the \(y\) axis. Name the axis of symmetry for: a) \(A(3,-4)\) and \(B(3,4)\) b) \(A(2,0)\) and \(B(-2,0)\) c) \(A(3,-4)\) and \(B(-3,-4)\) d) \(A(a, b)\) and \(B(a,-b)\)

Points \(A\) and \(B\) have symmetry with respect to the \(y\) axis. Find the coordinates of \(A\) if \(B\) is the point: a) \((3,-4)\) b) \((2,0)\) c) \((a, 0)\) d) \((b, c)\)
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