In the -plane, line has equation , where is a constant. Line is defined by . The midpoint of the segment that joins the origin to the intersection point of lines and lies on the line . What is the value of ?

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SAT Linear Equations in Two Variables Question 63 | Free SAT Prep | Aniko

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Question 63·Hard·Linear Equations in Two Variables

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In the $xy$ -plane, line $L$ has equation $y = \tfrac{2}{3}x + b$ , where $b$ is a constant. Line $M$ is defined by $4x + 6y = 9$ . The midpoint of the segment that joins the origin $(0,0)$ to the intersection point of lines $L$ and $M$ lies on the line $y = x + 1$ . What is the value of $b$ ?

For line-and-midpoint problems, first introduce variables for the unknown intersection point, then use the midpoint formula and any given line condition to create a new equation relating $x$ and $y$ . Combine that equation with one of the original line equations to solve for the intersection coordinates, and only at the end substitute that point into the remaining line equation to solve for the requested constant. This keeps the algebra organized and avoids trying to solve directly with three equations at once.

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Represent the intersection point and use the midpoint formula

Call the intersection point of lines $L$ and $M$ $(x,y)$ . What are the coordinates of the midpoint of the segment from $(0,0)$ to $(x,y)$ ?

Use the condition that the midpoint lies on $y = x + 1$

Take your midpoint $(x/2, y/2)$ and substitute it into $y = x + 1$ . What equation relating $x$ and $y$ do you get after clearing the fractions?

Solve for the intersection, then use line $L$

You now have two equations that the intersection point $(x,y)$ must satisfy: the equation from the midpoint condition and $4x + 6y = 9$ . Solve this system to find $(x,y)$ , then plug that point into $y = \tfrac{2}{3}x + b$ to solve for $b$ .

Desmos Guide

Graph the lines that determine the intersection point

Use the midpoint condition to note that the intersection point $(x,y)$ must satisfy $y = x + 2$ as well as $4x + 6y = 9$ . In Desmos, enter 4x+6y=9 and y=x+2. Click on their intersection to read off its coordinates $(x,y)$ .

Compute the value of $b$ from the intersection point

Using the intersection coordinates from Desmos, compute $b$ with the formula $b = y - \tfrac{2}{3}x$ . In Desmos, you can type something like y_value - (2/3)*x_value (replacing x_value and y_value with the numbers from the intersection point). Read the resulting decimal and convert it to a fraction to match it to one of the answer choices.

Step-by-step Explanation

Use the midpoint condition to relate $x$ and $y$

Let the intersection point of lines $L$ and $M$ be $(x,y)$ .

The segment is from the origin $(0,0)$ to $(x,y)$ .
The midpoint of this segment is
- $\left(\dfrac{0+x}{2},\dfrac{0+y}{2}\right) = \left(\dfrac{x}{2},\dfrac{y}{2}\right)$ .

We are told this midpoint lies on the line $y = x + 1$ . So plug the midpoint coordinates into that line:

\frac{y}{2} = \frac{x}{2} + 1.

Multiply both sides by $2$ to clear the denominator:

y = x + 2.

So the intersection point $(x,y)$ must also lie on the line $y = x + 2$ . This is an extra equation relating $x$ and $y$ .

Find the intersection point using line $M$ and $y=x+2$

The intersection point $(x,y)$ is on line $M$ , which has equation

4x + 6y = 9.

We also know $y = x + 2$ from the midpoint condition. Substitute $y = x + 2$ into $4x + 6y = 9$ :

4x + 6(x + 2) = 9 \\[0.7em] 4x + 6x + 12 = 9 \\[0.7em] 10x + 12 = 9 \\[0.7em] 10x = -3 \\[0.7em] x = -\frac{3}{10}.

Now find $y$ using $y = x + 2$ :

y = -\frac{3}{10} + 2 = -\frac{3}{10} + \frac{20}{10} = \frac{17}{10}.

So the intersection point of $L$ and $M$ is $\left(-\dfrac{3}{10},\dfrac{17}{10}\right)$ .

Use line $L$ to solve for $b$

Line $L$ has equation

y = \frac{2}{3}x + b.

The intersection point $\left(-\dfrac{3}{10},\dfrac{17}{10}\right)$ lies on line $L$ , so substitute $x=-\dfrac{3}{10}$ and $y=\dfrac{17}{10}$ into this equation:

\frac{17}{10} = \frac{2}{3}\left(-\frac{3}{10}\right) + b.

Simplify the product:

\frac{2}{3}\left(-\frac{3}{10}\right) = -\frac{6}{30} = -\frac{1}{5} = -\frac{2}{10}.

\frac{17}{10} = -\frac{2}{10} + b.

Add $\dfrac{2}{10}$ to both sides:

\frac{17}{10} + \frac{2}{10} = b \\[0.7em] \frac{19}{10} = b.

Thus, $b = \tfrac{19}{10}$ , which corresponds to choice B.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation