A line has a slope of -1/4 and includes the points (1, -3) and (-7, a). What is the value of a?

Understand the Problem

The question is asking us to find the value of 'a' in the coordinates of a line, given a specific slope and two points on that line. We will use the slope formula, which is (y2 - y1) / (x2 - x1), to determine the missing coordinate.

Answer

The value of $ a $ is $ -5 $.

Steps to Solve

Identify the known values We have the slope $m = -\frac{1}{4}$ and two points on the line:

Point 1: $(1, -3)$
Point 2: $(-7, a)$

Use the slope formula The slope formula is given by:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Here, we can substitute the known values into the formula where:

$y_1 = -3$
$y_2 = a$
$x_1 = 1$
$x_2 = -7$

Substitute the values into the formula Now we substitute the values into the slope formula:

$$ -\frac{1}{4} = \frac{a - (-3)}{-7 - 1} $$

This simplifies to:

$$ -\frac{1}{4} = \frac{a + 3}{-8} $$

Multiply both sides by -8 To eliminate the denominator, multiply both sides by -8:

$$ 8 \cdot \left(-\frac{1}{4}\right) = a + 3 $$

This gives:

$$ -2 = a + 3 $$

Solve for 'a' Now we solve for $a$ by isolating it:

$$ a = -2 - 3 $$

Thus,

$$ a = -5 $$

The value of ( a ) is ( -5 ).

More Information

The slope of a line indicates how steep it is and the direction it moves as you go along it. Knowing any two points on the line lets you calculate the slope and determine missing coordinates.

Tips

Forgetting to account for the negative sign in the slope calculation.
Mixing up the coordinates when applying the slope formula. Always ensure the correct ( (x_1, y_1) ) and ( (x_2, y_2) ) pairs are used.

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