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Complete the equation of the line through [tex]$(-6,-5)$[/tex] and [tex]$(-4,-4)$[/tex]. Use exact numbers.

[tex]\[ y = \][/tex]


Sagot :

To complete the equation of the line through the points [tex]\((-6, -5)\)[/tex] and [tex]\((-4, -4)\)[/tex], we will follow these steps:

1. Calculate the slope [tex]\( m \)[/tex]:

The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substituting the given points into the formula:
[tex]\[ m = \frac{-4 - (-5)}{-4 - (-6)} = \frac{-4 + 5}{-4 + 6} = \frac{1}{2} = 0.5 \][/tex]

2. Determine the y-intercept [tex]\( b \)[/tex]:

Use the slope-intercept form of the line equation, which is [tex]\( y = mx + b \)[/tex]. Substitute one of the points [tex]\((-6, -5)\)[/tex] into this equation along with the slope [tex]\( m = 0.5 \)[/tex]:
[tex]\[ -5 = 0.5 \cdot (-6) + b \][/tex]

Simplify to solve for [tex]\( b \)[/tex]:
[tex]\[ -5 = -3 + b \implies b = -5 + 3 \implies b = -2 \][/tex]

3. Write the equation of the line:

Now that we have the slope [tex]\( m = 0.5 \)[/tex] and the y-intercept [tex]\( b = -2 \)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[ y = 0.5x - 2 \][/tex]

So, the complete equation of the line is:
[tex]\[ y = 0.5x - 2 \][/tex]