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Chapter 3: Problem 29

Write the slope-intercept equation of the line that passes through the given points.$$(1,-5),(3,-11)$$

Short Answer

Expert verified
The slope-intercept equation of the line that passes through the points (1, -5) and (3, -11) is y = -3x - 2.

Step by step solution

01

Calculate the Slope (m)

To find the slope of the line, use the formula (y2 - y1) / (x2 - x1). Here, we have two points (1, -5) and (3, -11). We can refer to (1, -5) as (x1, y1) and (3, -11) as (x2, y2). Applying the formula, we calculate the slope (m) as (-11 - (-5)) / (3 - 1) = -6 / 2 = -3.
02

Substitute for m, x, and y in the Slope-Intercept Equation

After finding the slope (m), we can substitute m, x, and y from one of the given points into the slope-intercept form (y = mx + b). Let's use the point (1, -5) for x and y. This requires calculating -5 = -3*1 + b.
03

Calculate the Y - Intercept (b)

By performing the calculations in the equation from step 2, we can calculate b. This gives us -5 = -3 + b, or b = -5 + 3 = -2. Thus, our y-intercept (b) is -2.
04

Write the Equation of the Line in Slope-Intercept Form

Finally, we can write the equation of the line in slope-intercept form (y = mx + b) by substituting the slope (m) and the y-intercept (b) that we found in the previous steps. This gives us the equation y = -3x - 2.

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Key Concepts

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

Calculating Slope
When it comes to finding the equation of a line, calculating the slope is the first crucial step. The slope determines how steep a line is and its direction鈥攚hether it inclines upwards or downwards as it moves from left to right. To calculate the slope, use the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
This formula takes two points on the line,
  • \((x_1, y_1)\), the coordinates of the first point, and
  • \((x_2, y_2)\), the coordinates of the second point.
For example, if our points are \((1, -5)\) and \((3, -11)\), we plug these into the formula:
  • \( m = \frac{-11 - (-5)}{3 - 1} = \frac{-6}{2} = -3 \)
The slope \(m\) is found to be \(-3\), showing the line decreases sharply from left to right.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis on a graph. This is an essential part of the slope-intercept equation, which helps us describe the line completely.To find the y-intercept, we begin with the slope-intercept formula:
  • \( y = mx + b \)
Here, \(m\) is the slope, and \(b\) represents the y-intercept. After calculating the slope, substitute one of the given points into the formula to solve for \(b\). Using point \((1, -5)\):
  • \(-5 = -3 \cdot 1 + b \)
  • Simplifying gives us: \( b = -5 + 3 = -2 \)
Therefore, the y-intercept \(b\) is \(-2\). This calculation means the line touches the y-axis at \( -2 \).
Equation of a Line
Once we have both the slope and the y-intercept, we can write the equation of the line. This equation represents all points on that line and gives us a quick way to understand its behavior.Using the slope \(m = -3\) and y-intercept \(b = -2\), insert these into the slope-intercept form:
  • \( y = mx + b \)
  • \( y = -3x - 2 \)
This final equation, \(y = -3x - 2\), succinctly describes our line. It tells us for every step we move to the right by one unit, the line goes down by three units. Whenever \(x = 0\), the line will cross the y-axis at \(-2\). This is the complete picture of our line in mathematical form.
Linear Equations
Linear equations like \(y = mx + b\) are vital in mathematics because they describe straight lines, which are the simplest type of graph. These equations are used in various real-world applications, such as physics, economics, and everyday planning.Key aspects of linear equations include:
  • The slope \(m\) impacts how steep or flat the line is, determining the rate of change.
  • The y-intercept \(b\) tells us where the line crosses the y-axis, setting a baseline for the line's position.
A linear equation is easy to identify because it will always graph as a straight line and has no powers or roots of \(x\). Understanding the components of linear equations can help solve problems across a range of subjects effectively.

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Most popular questions from this chapter

Express each sentence as a formula. The current \(I\) varies directly with the voltage \(V\) and inversely with the resistance \(R\).

Set up a variation equation and solve for the requested value. The time it takes a car to travel a certain distance varies inversely with its rate of speed. If a certain trip takes 3 hours when the driver travels at \(50 \mathrm{mph}\), how long will the trip take when the driver travels at \(60 \mathrm{mph}\) ?

Solve each inequality and graph the solution. $$-5 x+4 \geq-6$$

Graph each equation using any method. $$y=4.5 x+2$$

Set up a variation equation and solve for the requested value. The interest earned on a fixed amount of money varies jointly with the annual interest rate and the time that the money is left on deposit. If an account earns \(120\)dollar at \(8 \%\) annual interest when left on deposit for 2 years, how much interest would be earned in 3 years at an annual rate of \(12 \% ?\)
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