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Chapter 2: Problem 44

Write an equation of the line satisfying the following conditions. Write the equation in the form \(y=\mathrm{mx}+b\). It passes through (7,-2) and its \(y\) -intercept is 5 .

Short Answer

Expert verified
The equation of the line is \(y=\frac{-1}{7}x + 5\).

Step by step solution

01

Identify the Given Points

From the problem, the given points are (7,-2) and (0,5). Here, (0,5) represents the y-intercept.
02

Calculate the Slope

The slope (m) of a line can be determined using the formula \[m = \frac{y2-y1}{x2-x1}\]. Now, by substituting the given points into this formula, the slope can be calculated.
03

Substitute the Slope and the Y-intercept into the Equation

Once the slope has been calculated, it can be substituted into the equation \(y=mx+b\), along with the y-intercept (b), to form the full equation of 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 of a linear equation is fundamentally the equation of a straight line. This form is expressed as \(y = mx + b\).
  • "\(y\)" represents the dependent variable, which depends on the value of \(x\).
  • "\(m\)" is the slope of the line. This shows how steep the line is.
  • "\(x\)" stands for the independent variable.
  • "\(b\)" is the y-intercept, or the point where the line crosses the y-axis.
In the equation \(y = mx + b\), each of these components helps define exactly what the line looks like on a graph. This form is widely used because it's easy to quickly identify the slope and y-intercept from the equation.
Y-intercept
The y-intercept is the point where the graph of a line intersects the y-axis. In mathematical terms, it is the value of \(y\) when \(x = 0\). As such, if you have a line in the form of \(y = mx + b\), "\(b\)" is your y-intercept.
  • For example, in the equation \(y = 2x + 5\), the y-intercept is \(5\).
  • This means that when \(x\) is zero, \(y = 5\).
  • On a graph, this would be depicted as the point \((0, 5)\).
When graphing a line, starting from the y-intercept makes the process easier. This fixed point, together with the slope, guides you in drawing the entire line.
Calculation of Slope
Calculating the slope of a line involves finding the change in \(y\) relative to the change in \(x\). This is often referred to as "rise over run."The general formula to calculate the slope \(m\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
  • Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
  • "\(y_2 - y_1\)" is the change in the y-direction (vertical change).
  • "\(x_2 - x_1\)" is the change in the x-direction (horizontal change).
For example, using points \((7, -2)\) and \((0, 5)\):\[m = \frac{5 - (-2)}{0 - 7} = \frac{7}{-7} = -1\]Thus, the slope \(m\) is \(-1\). This negative value indicates the line decreases as it moves from left to right.

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Most popular questions from this chapter

Write an equation of the line satisfying the following conditions. Write the equation in the form \(\mathrm{Ax}+\mathrm{By}=C\). It passes through (2,-3) and (5,1) .

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