In the xy-plane, line L_1 passes through the points (1, 1) and (3, 5). Line L_2 is defined… — GRE General Test

In the $xy$ -plane, line $L_1$ passes through the points $(1, 1)$ and $(3, 5)$ . Line $L_2$ is defined by the equation $\frac{x}{4} + \frac{y}{7} = 2$ . If $(x, y)$ is the point of intersection of lines $L_1$ and $L_2$ , what is the value of $x + y$ ?

11Answer
B
22
C
4
D
9.5
E
15

Answer

The value of

x + y

is 11.

The correct solution first determines the equation of the first line,

y = 2x - 1

, from its given points. Substituting this into the second line's equation

\frac{x}{4} + \frac{y}{7} = 2

and clearing fractions yields

x = 4

and

y = 7

. Adding these coordinates gives

4 + 7 = 11

Step-by-Step Solution

Determine the slope and equation of line

L_1

The slope

m = \frac{5 - 1}{3 - 1} = 2

. Using point-slope form with

(1, 1)

y - 1 = 2(x - 1)

, which simplifies to

y = 2x - 1

Two points uniquely define a line, allowing us to express

y

in terms of

x

Substitute the expression for

y

into the equation for line

L_2

\frac{x}{4} + \frac{2x - 1}{7} = 2

At the intersection point, both equations share the exact same

(x, y)

values.

Clear the denominators by multiplying the equation by the least common multiple, 28.

7x + 4(2x - 1) = 56 \implies 7x + 8x - 4 = 56 \implies 15x = 60 \implies x = 4

Eliminating fractions simplifies solving for

x

Calculate the value of

y

and find

x + y

y = 2(4) - 1 = 7

, so

x + y = 4 + 7 = 11

The question specifically asks for the sum of the intersection coordinates.

Key Concept

Systems of Linear Equations and Line Intersections

Estimated Time:1m 30s

Question

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