In the xy-plane, the graph of a line l passes through the points (0, 1) and (3, 5). If ano… — SAT

In the $xy$ -plane, the graph of a line $l$ passes through the points $(0, 1)$ and $(3, 5)$ . If another point on line $l$ has coordinates $(t, 9)$ , what is the value of $t$ ?

Answer: 6

Answer

The value of

t

6

The slope of line

l

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

. Using the y-intercept

(0, 1)

, the equation of the line is

y = \frac{4}{3}x + 1

. Setting

y = 9

gives

9 = \frac{4}{3}t + 1

. Subtracting 1 from both sides gives

8 = \frac{4}{3}t

. Multiplying both sides by

\frac{3}{4}

yields

t = 6

Step-by-Step Solution

Calculate the slope of line

l

using the slope formula

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

with the points

(0, 1)

and

(3, 5)

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

The slope of a line represents its constant rate of change and is needed to determine the line's equation.

Write the equation of the line in slope-intercept form,

y = mx + b

, using the slope

m = \frac{4}{3}

and the y-intercept

b = 1

(from the point

(0, 1)

y = \frac{4}{3}x + 1

The slope-intercept equation defines the relationship between the

x

- and

y

-coordinates of any point on the line.

Substitute the point

(t, 9)

into the line's equation and solve for

t

9 = \frac{4}{3}t + 1 \implies 8 = \frac{4}{3}t \implies t = 6

Since the point lies on the line, its coordinates must satisfy the line's equation.

Key Concept

Determining the equation of a linear function from a graph or points and evaluating it for a given value.

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