Question 96·Hard·Systems of Two Linear Equations in Two Variables
In the system of equations below, is a constant.
For what value of does the system have an infinite number of solutions ?
For SAT questions asking when a system has infinitely many solutions, remember that both equations must represent the same line. Quickly compare coefficients: find the factor that takes one equation’s -coefficient and constant to the other’s, then apply that same factor to the -coefficient to form a simple equation in the parameter (here, ). Solve that one-step equation and avoid guessing based only on matching a single coefficient or the constants.
Hints
Think about what "infinitely many solutions" means for lines
When do two linear equations in and have infinitely many solutions? Think about how their graphs must look relative to each other.
Compare corresponding parts of the two equations
Look at the coefficients of and the constant terms in both equations. How does become ? How does become ? What does that tell you about a multiplier relating the two equations?
Use the same multiplier for the -terms
Once you know the factor that takes to , apply that same factor to the -term . What equation must that give you so it matches the in the second equation?
Desmos Guide
Enter the equations with a slider for k
Type k in Desmos to create a slider, then enter the first equation as k*x + 3y = 7 and the second equation as 4x + 12y = 28. You should see two lines on the graph and a slider for .
Adjust k to see when the lines coincide
Move the -slider slowly and watch how the line for moves. Look for the value of at which this line lies exactly on top of the line so they are indistinguishable.
Read the k-value for infinitely many solutions
When the two lines perfectly overlap, the system has infinitely many solutions. Read the corresponding -value from the slider at that moment.
Step-by-step Explanation
Recall the condition for infinitely many solutions
A system of two linear equations in and has:
- One solution when the lines intersect once.
- No solution when the lines are parallel but different.
- Infinitely many solutions when both equations represent the exact same line.
For the equations to represent the same line, all corresponding coefficients (for , , and the constant term) must be in the same ratio.
Compare the -coefficients and constants to find the scale factor
Start with the system:
If the second equation is a multiple of the first, there is some number such that:
should match
Compare the -coefficients and constants:
- From to , the factor is such that .
- From to , the factor is also such that .
Solve one of these for :
(You can check also gives .)
Use the same factor to relate the -coefficients
Since the same factor must work for all parts of the equation, the -coefficients must also match under .
The -term of the first equation is . After multiplying by , it becomes .
This must equal the -term of the second equation, which is . So we get the equation
Solve for the value of
Solve the equation from the previous step:
So the system has an infinite number of solutions when .