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The line crosses through the points [tex]\((-7, -5)\)[/tex], [tex]\((0, 1)\)[/tex], and [tex]\((7, 7)\)[/tex]. Write the equation of the line in slope-intercept form.

Sagot :

Certainly! Let's find the equation of the line that passes through the points \((-7, -5)\), \((0, 1)\), and \((7, 7)\). The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

### Step 1: Calculate the Slope (\(m\))

The slope \(m\) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Using the points \((-7, -5)\) and \((0, 1)\):
[tex]\[ x_1 = -7, \quad y_1 = -5 \][/tex]
[tex]\[ x_2 = 0, \quad y_2 = 1 \][/tex]

Substitute these values into the slope formula:
[tex]\[ m = \frac{1 - (-5)}{0 - (-7)} = \frac{1 + 5}{0 + 7} = \frac{6}{7} \][/tex]

So, the slope \(m\) is:
[tex]\[ m \approx 0.857 \][/tex]

### Step 2: Calculate the y-intercept (\(b\))

The y-intercept \(b\) can be found using the slope-intercept equation \(y = mx + b\). We can use any of the given points to find \(b\); let’s use the point \((0, 1)\).

Substitute \(x = 0\) and \(y = 1\) into the equation \(y = mx + b\):

[tex]\[ 1 = 0.857 \cdot 0 + b \][/tex]

Since \(0.857 \cdot 0 = 0\), we get:
[tex]\[ b = 1 \][/tex]

### Step 3: Write the Equation of the Line

Now we have found both the slope (\(m\)) and the y-intercept (\(b\)). Thus, the equation of the line in slope-intercept form is:

[tex]\[ y = 0.857x + 1 \][/tex]

So, the equation of the line that passes through the points \((-7, -5)\), \((0, 1)\), and \((7, 7)\) is:
[tex]\[ y = 0.857 \, x + 1 \][/tex]

This gives us a linear equation that accurately represents the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] for the line passing through the given points.