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Chapter 5: Problem 52

Write the slope-intercept form of the equation of the line, if possible, given the following information. contains \((-6,-5)\) and \((4,10)\)

Short Answer

Expert verified
The equation of the line in slope-intercept form is: \(y = \frac{3}{2}x + 4\).

Step by step solution

01

Find the slope of the line

Using the given points \((-6, -5)\) and \((4, 10)\), we can find the slope (m) using the following formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Substitute the values into the formula: \(m = \frac{10 - (-5)}{4 - (-6)}\)
02

Simplify the slope

Simplify the expression to find the slope: \(m = \frac{10 + 5}{4 + 6} = \frac{15}{10}\) Divide the numerator and the denominator by the greatest common divisor, 5: \(m = \frac{15 梅 5}{10 梅 5} = \frac{3}{2}\) So, the slope of the line is \(\frac{3}{2}\).
03

Find the y-intercept (b)

Using the slope (\(\frac{3}{2}\)) and one of the points (\((-6, -5)\)), we can plug the values into the equation: \(y = mx + b\) Substitute the values: \(-5 = \frac{3}{2}(-6) + b\)
04

Solve for b

Now, we need to solve the equation to find the y-intercept (b): \(-5 = -9 + b\) Add 9 to both sides: \(b = -5 + 9\) Simplify: \(b = 4\) So, the y-intercept is 4.
05

Write the equation in slope-intercept form

Finally, we have the slope (\(\frac{3}{2}\)) and the y-intercept (4). Now, we can write the equation of the line in slope-intercept form: \(y = mx + b\) Substitute the values: \(y = \frac{3}{2}x + 4\) So, the equation of the line in slope-intercept form is: \(y = \frac{3}{2}x + 4\).

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Key Concepts

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

Slope Calculation
The slope of a line tells us how steep the line is and the direction it is going. To determine the slope between two points, you can use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula uses two points on the line, labeled as \((x_1, y_1)\) and \((x_2, y_2)\).Let's break it down:
  • Subtract the y-values: \( y_2 - y_1 \).
  • Subtract the x-values: \( x_2 - x_1 \).
  • Divide the difference in y-values by the difference in x-values.
The result is the slope \(m\), which tells us how much the y-value changes for each unit increase in the x-value.
In our example, using the points \((-6, -5)\) and \((4, 10)\), we find the slope is \( \frac{3}{2} \). This means the line rises 3 units for every 2 units it runs to the right.
Y-intercept Determination
The y-intercept is the point where the line crosses the y-axis. It is determined when the x-coordinate is zero. In slope-intercept form, the y-intercept is represented by \(b\) in the equation \(y = mx + b\).Here's how to determine the y-intercept:
  • Start with the slope value that you already calculated, \( \frac{3}{2} \) in our example.
  • Use one of the given points, like \((-6, -5)\).
  • Substitute these values into the equation \(y = mx + b\).
Next, solve for \(b\):
  • Plug in \(y = -5\) and \(x = -6\) into the equation: \(-5 = \frac{3}{2}(-6) + b\).
  • Calculate \(\frac{3}{2} \times (-6) = -9\), and solve for \(b\).
  • Add 9 to both sides to get \(b = 4\).
The y-intercept, \(b\), is thus 4. This means the line crosses the y-axis at the point (0, 4).
Equation of a Line
The equation of a line in slope-intercept form is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form makes it easy to read off the slope and y-intercept directly from the equation.With our calculated slope (\( \frac{3}{2} \)) and y-intercept (4), the equation of our line becomes:
  • Start with the slope-intercept form: \( y = mx + b \).
  • Substitute \( m = \frac{3}{2} \) and \( b = 4 \) into the formula.
  • The equation becomes: \( y = \frac{3}{2}x + 4 \).
This equation describes the line passing through the given points \((-6, -5)\) and \((4, 10)\). By simply inputting different values of \(x\), you can find corresponding values of \(y\) on this line, illustrating its linear nature.

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