A line has a slope of 1 and includes the points (3, 4) and (-4, k). What is the value of k?
Understand the Problem
The question asks for the value of k in a line with a slope of 1, given two points: (3, 4) and (-4, k). To solve it, we will use the slope formula, which is (y2 - y1) / (x2 - x1).
Answer
The value of \( k \) is \( -3 \).
Answer for screen readers
The value of ( k ) is ( -3 ).
Steps to Solve
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Identify the points The two points are given as (3, 4) and (-4, k).
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Use the slope formula The slope formula is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Here, the slope ( m ) is 1. Let’s label the points:
- Point 1: ( (x_1, y_1) = (3, 4) )
- Point 2: ( (x_2, y_2) = (-4, k) )
- Plug values into the slope equation Substituting the known values:
$$ 1 = \frac{k - 4}{-4 - 3} $$
- Simplify the equation Calculate the denominator:
$$ -4 - 3 = -7 $$
So, the equation now is:
$$ 1 = \frac{k - 4}{-7} $$
- Clear the fraction Multiply both sides by -7:
$$ -7 = k - 4 $$
- Solve for ( k ) Add 4 to both sides:
$$ k = -7 + 4 $$
This simplifies to:
$$ k = -3 $$
The value of ( k ) is ( -3 ).
More Information
The slope of a line indicates its steepness and direction. In this case, since the slope is 1, the line rises one unit vertically for every one unit it moves horizontally. The calculation shows how the coordinates are related through the slope formula.
Tips
- Forgetting to change the sign in the slope formula when substituting ( x_2 ) and ( x_1 ).
- Not properly isolating ( k ) after clearing the fraction.
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