A line has a slope of 10/3 and includes the points (-4, t) and (-1, 9). What is the value of t?

Understand the Problem

The question is asking for the value of the variable t, given the slope of a line and two points that lie on the line. To solve it, we can use the formula for the slope between two points and rearrange it to find t.

Answer

The value of $ t $ is $ -1 $.

Steps to Solve

Identify the slope formula
The formula for the slope ( m ) between two points ((x_1, y_1)) and ((x_2, y_2)) is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Assign values to the points
In this problem, we have two points:

Point 1: ((-4, t)) where (x_1 = -4) and (y_1 = t)
Point 2: ((-1, 9)) where (x_2 = -1) and (y_2 = 9)

Substitute the values into the slope formula
We know the slope ( m = \frac{10}{3} ). Therefore, we can set up the equation:
$$ \frac{9 - t}{-1 - (-4)} = \frac{10}{3} $$
Simplify the expression
The denominator simplifies as follows:
$$ -1 - (-4) = -1 + 4 = 3 $$
Now the equation becomes:
$$ \frac{9 - t}{3} = \frac{10}{3} $$
Clear the fraction by multiplying both sides by 3
$$ 9 - t = 10 $$
Solve for ( t )
Subtract 9 from both sides:
$$ -t = 10 - 9 $$
$$ -t = 1 $$
Now, multiply by -1:
$$ t = -1 $$

The value of ( t ) is ( -1 ).

More Information

Finding the length or output of a segment is essential in determining line equations and graphical representation in mathematics. Slope helps in defining the angle of inclination of the line and characterizes its direction.

Tips

Forgetting to simplify the denominator properly.
Mixing up the coordinates ( (x_1, y_1) ) and ( (x_2, y_2) ).
Not applying the correct sign when isolating the variable.

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