Question 138·Hard·Systems of Two Linear Equations in Two Variables
In the system of equations above, is a nonzero constant. For what value of does the system have infinitely many solutions?
(Express the answer as an integer)
For SAT system-of-equations questions asking about infinitely many solutions, immediately think: the two equations must represent the same line, so one equation must be a constant multiple of the other. Simplify the equations (by dividing out common factors) and factor out any parameters like so you can clearly see common expressions (such as here). Then set up an equation that forces the second equation to be a multiple of the first (often by substitution or by matching coefficient ratios) and solve for the parameter, instead of solving for and directly.
Hints
Simplify the equations
Try to make both equations look similar: divide the first equation by , and see if you can factor anything out of the second equation.
Look for a common expression
After simplifying, you should see the expression appear in both equations. Rewrite the second equation using this common expression.
Use substitution with
From the first equation, you can find what equals. Substitute that value into the second equation and solve for so that both equations can be true at the same time for many pairs.
Think about 'infinitely many solutions'
Two linear equations in and have infinitely many solutions only when they represent the exact same line. That means one equation must be a constant multiple of the other.
Desmos Guide
Enter the equations
In Desmos, type the first equation as 4x + 2y = 8. Then type the second equation as 2k x + k y = 12. Desmos will automatically create a slider for the parameter k.
Use the slider to compare the lines
Adjust the slider for k and watch how the second line (from 2k x + k y = 12) moves on the graph compared to the first line. You are looking for the value of k where the two lines lie exactly on top of each other.
Identify the required k-value
When the two lines completely overlap (they are indistinguishable on the graph), note the current value of k shown on the slider. That is the value of k for which the system has infinitely many solutions.
Step-by-step Explanation
Simplify the first equation
Start by dividing every term of the first equation by :
So the first equation can be written as .
Factor the second equation
In the second equation, factor out from the left side:
So the second equation can be written as .
Apply the condition for infinitely many solutions
For the system to have infinitely many solutions, every solution of the first equation must also satisfy the second equation.
From the first equation, we know that for all solutions, .
Substitute into the second equation :
which simplifies to the equation
Solve for k
Solve the equation from the previous step:
So the value of that makes the system have infinitely many solutions is .