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Chapter 4: Problem 438

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (0,4) and (2,-3)

Short Answer

Expert verified
The equation of the line is \( y = -\frac{7}{2}x + 4 \).

Step by step solution

01

- Find the Slope

To find the slope of the line passing through the points (0, 4) and (2, -3), use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) Substitute the given points into the formula: \( m = \frac{-3 - 4}{2 - 0} = \frac{-7}{2} \). The slope of the line is \( m = -\frac{7}{2} \).
02

- Use the Slope-Intercept Form

The slope-intercept form of a line's equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the point (0, 4), it is clear that the y-intercept \( b \) is 4 since this is the y-value where the line crosses the y-axis.
03

- Write the Equation

Now that the slope \( m \) and y-intercept \( b \) are known, substitute them into the slope-intercept form equation: \( y = -\frac{7}{2}x + 4 \). This is the equation of the line in slope-intercept form.

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Key Concepts

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

Find the Slope
To determine the slope of a line, we use the slope formula. The slope gives us the steepness and direction of the line. Imagine the line connects two points: (0, 4) and (2, -3). We can use the slope formula: \( m = \frac{ y_2 - y_1 }{ x_2 - x_1 } \) Here, \( y_1 \) and \( y_2 \) are the y-coordinates of the points, while \( x_1 \) and \( x_2 \) are the x-coordinates. Substituting the given points:\( m = \frac{ -3 - 4 }{ 2 - 0 } = \frac{ -7 }{ 2 } \)This simplifies to \( m = -\frac{ 7 }{ 2 } \). Therefore, the slope of the line is \( -\frac{ 7 }{ 2 } \). This means the line is declining as we move from left to right.
Slope Formula
The slope formula is essential in determining how a line inclines or declines between two points. Here鈥檚 a closer look at what it involves:When given two points \(( x_1, y_1 )\) and \(( x_2, y_2 )\): \( \text{Slope} ( m ) = \frac{ \text{Difference in y-values} }{ \text{Difference in x-values} } \)It essentially measures how much the y-coordinates (vertical change) change as the x-coordinates (horizontal change) change. The result \( m \) tells you whether the line ascends (positive slope), descends (negative slope), or is level (slope of zero). Below are a few tips to remember:
  • Positive slope: Line rises from left to right.
  • Negative slope: Line falls from left to right.
  • Zero slope: Line is horizontal.
  • Undefined slope: Line is vertical.
The slope allows us to understand the rate at which the variables change relative to each other.
Y-Intercept
The y-intercept is where the line crosses the y-axis. In the equation format \( y = mx + b \), the聽y-intercept is denoted by \( b \). It represents the value of y when x is zero. Given the points (0, 4) and (2, -3), it鈥檚 clear that at x = 0, y = 4. Hence, our y-intercept \( b \) is 4.When interpreting graph data:
  • The y-intercept tells you the starting value when x is zero.
  • It helps in plotting the initial point of the graph.
  • In real-life scenarios, it could represent a fixed initial quantity.
In our example, the equation of the line becomes clearer: Once we know \( b \) and the slope, we can finalize our line equation.
Equation of a Line
Putting everything together, the slope-intercept form of the equation of a line is \( y = mx + b \). This format is very useful because it directly shows the slope \( m \) and the y-intercept \( b \). For our example, we already found that the slope \( m \) is \( -\frac{ 7 }{ 2 } \) and the y-intercept \( b \) is 4.Using these components, we can now substitute back into the slope-intercept form:\( y = -\frac{ 7 }{ 2 } x + 4 \)This represents the complete equation of the line that passes through points (0, 4) and (2, -3). To summarize:
  • The slope helps us determine the line鈥檚 direction and steepness.
  • The y-intercept lets us know exactly where the line will cut through the y-axis.
  • The equation incorporates both these elements to precisely map the line.
Understanding these relationships is crucial for graphing lines and predicting their behavior.

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Most popular questions from this chapter

Graph the linear inequality \(y \geq \frac{3}{2} x\)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(2 x+5 y=6,\) point(0,0)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(3 x-y=4,\) point (3,1)

Find the equation of each line. Write the equation in slope-intercept form. \(m=\frac{1}{6},\) containing point (6,1)

Graph the linear inequality \(y \leq-x\)
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