Question 101·Medium·Systems of Two Linear Equations in Two Variables
Consider the system of linear equations.
For what value of does this system have no solution?
For SAT questions where a system with a parameter (like ) is said to have no solution or infinitely many solutions, immediately think in terms of the lines’ slopes and intercepts. Put each equation into form so you can compare slopes quickly: equal slopes with different intercepts mean no solution (parallel lines), while equal slopes and equal intercepts mean infinitely many solutions (same line). If the algebra looks messy, you can also use elimination: choose a variable to eliminate, and if you end up with an equation like , you’ve found the parameter value that gives no solution.
Hints
Think about the graphs of the equations
If a system of two linear equations has no solution, what must the two lines look like on a graph? Think about their slopes and whether they ever intersect.
Put the equations into y = mx + b form
Rewrite each equation so it looks like . This will let you read off the slope for each line. One of the slopes will involve .
Match slopes, then check intercepts
Set the two slopes equal to each other and solve for . After you find , make sure the -intercepts are not the same so the lines are parallel but not identical.
Desmos Guide
Graph the first equation
In Desmos, type the first equation exactly as given: 2x - 3y = 17. Desmos will graph this line.
Test each answer choice for the second equation
For each option , type a corresponding second equation into Desmos, like -8x + 6y = 2, -4x + 6y = 2, 4x + 6y = 2, or 8x + 6y = 2. You can leave several of them on the screen at once, each as a separate line.
Look for the case with no intersection
Observe the graph for each value of . For most values, the second line will cross the first line at a point (one solution). Identify the value of for which the second line is parallel to the first line and never intersects it—this is the correct choice.
Step-by-step Explanation
Understand what “no solution” means
A system of two linear equations in and has no solution when the two lines are parallel and distinct.
- Parallel lines have the same slope.
- Distinct lines have different -intercepts (or, more generally, not all coefficients including the constant term are in the same ratio).
So we need to find the value of that makes the two lines have the same slope but not be the exact same line.
Write each equation in slope-intercept form
Start with the first equation:
Solve for :
So the slope of the first line is .
Now take the second equation:
Solve for :
So the slope of the second line is .
Set the slopes equal to make the lines parallel
For the system to have no solution, the slopes must match (lines parallel) but the intercepts must differ.
Set the slopes equal:
Now clear the denominators by cross-multiplying:
which simplifies to an equation relating :
We will solve this for in the next step.
Solve for k and confirm the lines are distinct
From the equation
divide both sides by :
Check that the lines are distinct: the first line has -intercept , and the second line has -intercept , which are not equal. So the lines are parallel and different, meaning the system has no solution when .