In the system of equations below, is a constant. The two lines intersect at the point . If , what is the value of ?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 87 | Free SAT Prep | Aniko

00:00

Question 87·Hard·Linear Equations in Two Variables

0 / 140

In the system of equations below, $k$ is a constant.

\begin{aligned} 2x + ky &= 6 \\ 3x - 4y &= 5 \end{aligned}

The two lines intersect at the point $(p,q)$ . If $p - q = 4$ , what is the value of $k$ ?

When a system problem gives you information about the intersection point, first translate that information into an extra equation in $x$ and $y$ (here, $p - q = 4$ becomes $x - y = 4$ ). Use this simpler relation together with one of the original equations that does not contain the parameter (here, $3x - 4y = 5$ ) to find the actual intersection coordinates. Finally, substitute that point into the remaining equation that contains the parameter and solve directly for the parameter. This avoids messy algebra with $k$ in multiple places and keeps the work fast and organized.

Hints

Click me!

Relate $p$ and $q$ to $x$ and $y$

The intersection point of the two lines is $(p,q)$ , and at that same point the coordinates are also $(x,y)$ . How can you rewrite the condition $p - q = 4$ as an equation involving $x$ and $y$ ?

Use substitution with the simpler equation

Once you have an equation like $x - y = 4$ , solve it for one variable (for example, $x$ in terms of $y$ ) and substitute into the equation $3x - 4y = 5$ to find the exact $x$ and $y$ values.

Use the first equation only to isolate $k$

After you know the intersection point $(x,y)$ , plug those values into $2x + ky = 6$ . Then solve that single equation for $k$ by isolating $k$ on one side.

Desmos Guide

Graph the two equations that do not involve $k$

In Desmos, enter the equations $3x - 4y = 5$ and $x - y = 4$ . These represent the second original line and the equation you get from the condition $p - q = 4$ .

Find the intersection point

Look at the graph and click on the point where the lines $3x - 4y = 5$ and $x - y = 4$ intersect. Note the $x$ - and $y$ -coordinates of this intersection point; they are the values of $x$ and $y$ at which both equations are true.

Use the intersection coordinates to compute $k$

In a new Desmos expression line, type $2x + ky = 6$ but replace $x$ and $y$ with the numeric coordinates you found (for example, type 2*(x_value) + k*(y_value) = 6). Then solve this equation for $k$ using Desmos (for instance, by using the solve command or by isolating $k$ algebraically and evaluating the expression). The resulting numeric value shown by Desmos is the required $k$ .

Step-by-step Explanation

Turn the condition on $(p,q)$ into an equation with $x$ and $y$

The intersection point of the two lines is $(p,q)$ , and we are told that $p-q=4$ .

At the intersection, the coordinates are also $(x,y)$ that satisfy both line equations, so we can write:

x - y = 4

Now solve this for $x$ :

x = y + 4

This expression will be useful to substitute into one of the given equations.

Use the second equation to find the actual intersection point

Substitute $x = y + 4$ into the second equation $3x - 4y = 5$ :

3(y+4) - 4y = 5

Simplify:

\begin{aligned} 3y + 12 - 4y &= 5 \\ -y + 12 &= 5 \end{aligned}

Solve for $y$ :

\begin{aligned} -y &= 5 - 12 = -7 \\ y &= 7 \end{aligned}

Now use $x = y + 4$ to find $x$ :

x = 7 + 4 = 11

So the intersection point is $(x,y) = (11,7)$ .

Substitute the intersection point into the first equation to solve for $k$

Now use the first equation $2x + ky = 6$ and plug in $x = 11$ and $y = 7$ :

2(11) + k(7) = 6

Simplify:

22 + 7k = 6

Solve for $k$ :

\begin{aligned} 7k &= 6 - 22 = -16 \\ k &= \frac{-16}{7} \end{aligned}

Thus, the value of $k$ is $-\dfrac{16}{7}$ , which corresponds to choice B.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation