Questions: Write an equation for the line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. (1,3) and (2,4) (a) The equation of the line in slope-intercept form is

Write an equation for the line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. (1,3) and (2,4) (a) The equation of the line in slope-intercept form is
Transcript text: Write an equation for the line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. \[ (1,3) \text { and }(2,4) \] (a) The equation of the line in slope-intercept form is $\square$
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Solution

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Solution Steps

To find the equation of the line passing through two given points, we need to:

  1. Calculate the slope (m) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  2. Use the slope and one of the points to find the y-intercept (b) using the formula \( y = mx + b \).
  3. Write the equation in slope-intercept form \( y = mx + b \).
Step 1: Calculate the Slope

To find the slope \( m \) of the line passing through the points \((1, 3)\) and \((2, 4)\), we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points: \[ m = \frac{4 - 3}{2 - 1} = \frac{1}{1} = 1.0 \]

Step 2: Calculate the Y-Intercept

Using the slope \( m = 1.0 \) and one of the points, say \((1, 3)\), we find the y-intercept \( b \) using the equation of the line \( y = mx + b \): \[ 3 = 1.0 \cdot 1 + b \implies b = 3 - 1.0 = 2.0 \]

Step 3: Write the Equation in Slope-Intercept Form

Now that we have the slope \( m = 1.0 \) and the y-intercept \( b = 2.0 \), we can write the equation of the line in slope-intercept form: \[ y = 1.0x + 2.0 \]

Final Answer

\[ \boxed{y = x + 2} \]

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