Two lines either cross once, never meet, or sit on top of each other. The SAT hides which one inside a single missing constant.
What is the value of k?
3x + ky = 12 6x + 10y = 7
In the given system of equations, k is a constant. If the system has no solution, what is the value of k?
No solution means the lines are parallel, so the x and y coefficients must be in the same ratio: 36 = k10, giving 6k = 30 and k = 5. The constants must then disagree, and they do — 3x + 5y = 12 doubled is 6x + 10y = 24, not 7 — so the lines really are parallel rather than identical. A stops at the scale factor: 6 is twice 3, and 2 is that multiplier, not the coefficient it produces. C copies the 10 from the matching slot, as if “same slope” meant “same number”. D sets the proportion upside down, 36 = 10k, which gives 20.
1
Two straight lines can do exactly three things.
Cross once, never meet, or lie on top of one another. There is no fourth picture and no system with exactly two solutions.
different slopes
exactly one solution
they lean differently, so they must cross
same slope, different intercept
no solution
parallel — a fixed gap that never closes
same slope, same intercept
infinitely many
one line wearing two disguises
“No solution” is not a mistake and does not mean you failed to solve it. It is a fact about the pair of equations: no ordered pair (x, y) in the xy-plane makes both of them true at once.
2
No solution: match the slopes, then check the constants disagree.
Back to the opening system. You never need the slopes themselves — matching the coefficients does the same job.
3x + ky = 12 and 6x + 10y = 7
both already in standard form ax + by = c
36 = k10
x-coefficients in the same ratio as the y-coefficients
6k = 3 · 10 = 30
cross-multiply
k = 5
divide both sides by 6
3x + 5y = 12 → ×2 → 6x + 10y = 24
and 24 ≠ 7, so the lines are distinct
k = 5 — same slope, different constant, so the lines never meet
Half the work is the second half
Matching the coefficients only forces the lines to be parallel or identical. The constant check is what rules out identical. Skip it and you will answer “no solution” to an item whose real answer is “infinitely many” — the difference between choice B and choice D on this kind of question.
3
Infinitely many: one common ratio, constants included.
If one whole equation is a multiple of the other — every coefficient and the number on the right — they are the same line, and every point on it solves the system.
4x − 6y = 10 and 10x − 15y = c
c is a constant; find c for infinitely many solutions
410 = 25 and −6−15 = 25
the coefficients already share one ratio
the multiplier is 10 ÷ 4 = 2.5
the second equation is 2.5 times the first
c = 2.5 × 10 = 25
the constant has to ride along
4x − 6y = 10 → ×2.5 → 10x − 15y = 25 ✓
one line, written twice
no solution → coefficients match, constant does notinfinitely many → everything matches
4
One test from standard form answers all three cases.
Write both as a₁x + b₁y = c₁ and a₂x + b₂y = c₂, then compare two ratios against a third.
compare a₁a₂ · b₁b₂ · c₁c₂
What you find
Picture
Solutions
a₁a₂ ≠ b₁b₂
lines cross
exactly one
a₁a₂ = b₁b₂ ≠ c₁c₂
parallel
none
a₁a₂ = b₁b₂ = c₁c₂
the same line
infinitely many
Line the equations up first
The test only works when x, y and the constant sit in the same places in both equations. Turn y = 4x + 3 into 4x − y = −3 before you compare anything, and keep the signs: the y-coefficient of 4x − y = −3 is −1, not 1. If a stem gives you one equation in slope form and one in standard form, that mismatch is the whole difficulty of the item.
5
What parallel looks like, and what algebra says about it.
Here is x − 3y = −9 with 2x − 6y = 18. The coefficients are in the ratio 12 and −3−6 = 12, but the constants give −918 = −12. Same slope, different intercept — and the vertical gap between them is 6 at every single x.
x − 3y = −9 → ×2 → 2x − 6y = −18
make the x and y terms identical
2x − 6y = −18 and 2x − 6y = 18
the same expression, two different values
subtracting gives 0 = −36
false for every x and y → no solution
in slope form: y = x3 + 3 and y = x3 − 3
same slope 13, intercepts 6 apart — the picture above
had the second been 2x − 6y = −18, subtracting gives 0 = 0
true for every point on the line → infinitely many
Desmos settles it in ten seconds
The Digital SAT gives you the graphing calculator throughout. Type both equations in exactly as printed — Desmos accepts standard form — and look: two separate lines that stay apart means no solution, and a single line that never thickens means the two equations drew the same thing, so infinitely many. For a system with a letter in it, add a slider for that letter and drag until the lines lock together.
6
Where the letter sits decides what the question can ask.
A constant in a coefficient and a constant on the right-hand side behave completely differently, and the SAT words the item to match.
7x − 3y = 9 and 21x + ky = 4
the letter is a coefficient — “for what value of k is there no solution?”
721 = −3k
force the slopes to agree
7k = −63, so k = −9
and 94 ≠ 13, so the lines stay apart ✓
Now move the letter to the right-hand side and the same system tells a different story.
7x − 3y = 9 and 21x − 9y = c
the slopes already match, whatever c is
7x − 3y = 9 → ×3 → 21x − 9y = 27
c = 27 makes them one line: infinitely many
every other c → no solution
c = 0, c = 26, c = 1,000 all give parallel lines
“Which is NOT a possible value?”
That is why an item with the letter on the right never asks “for what value of c is there no solution” — there are infinitely many such values. It asks for the infinitely many solutions value, or it flips to “the system has no solution; which of the following is NOT a possible value of c?” Both questions have the same answer: the one c that makes the two equations identical.
4x − 6y = 10 10x − 15y = 2525 = 25 = 25one line → infinitely many ✓
x − 3y = −9 2x − 6y = 180 = −36 on eliminationa false statement → no solution ✓
Remember the flow
Standard form · a₁a₂ = b₁b₂ · then ask what c₁c₂ does
Coefficients alone give parallel-or-identical. The constants break the tie.
Now you try
Practice problem
Which value is NOT possible?
y = 5x − 4 10x − 2y = d
In the given system of equations, d is a constant. If the system has no solution, which of the following is NOT a possible value of d?
Put the first equation in standard form: y = 5x − 4 becomes 5x − y = 4, and doubling it gives 10x − 2y = 8. So d = 8 makes the two equations the same line — infinitely many solutions, not none. Every other value of d leaves the slopes matched and the constants apart, which is exactly parallel, so 8 is the one value that cannot occur. C is 4, the constant before the doubling — the value you get by forgetting that multiplying 5x − y = 4 by 2 multiplies the 4 as well. A is −8, the same slip with a sign flip from moving the −4 across. B is 0, which is a perfectly ordinary value here: 10x − 2y = 0 is parallel to the first line and the system has no solution.
Hint: the question is upside down on purpose. Find the value of d that gives infinitely many solutions first — that single value is the one “no solution” cannot use, and every other choice is fine.
