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SAT Systems With Infinitely Many Solutions

This is the twin of the no-solution question. You get two linear equations, at least one unknown constant, and a question asking which value produces infinitely many solutions. The answer comes from recognizing that the two equations have to describe the exact same line, which is a stronger condition than parallel.

The rule

A system has infinitely many solutions when one equation is a constant multiple of the other, including the number on the right side.

Infinitely many solutions means every point on one line is also on the other, so the two equations are the same line written two different ways. That happens only when one entire equation is a constant multiple of the other. The x coefficient, the y coefficient, and the number on the right all have to scale by the same factor.

Take the system 2x plus 5y equals 7 and 6x plus ay equals b. Compare the x coefficients first, because they are the pair with no unknowns. Going from 2 to 6 means multiplying by 3. So the whole first equation multiplied by 3 gives 6x plus 15y equals 21. Matching term by term, a must be 15 and b must be 21. If the question only asks for a, you still found the multiplier the same way.

Some versions give you the multiplier on the other side. If the system is 4x minus y equals 3 and cx minus 2y equals 6, notice the right sides first: 3 to 6 is a factor of 2. Then the y terms confirm it, since negative 1 times 2 is negative 2. So c must be 4 times 2, which is 8. Any of the three pairs can be your anchor. Use whichever pair has no unknown in it.

Writing both equations in slope-intercept form also works and is worth doing when the coefficients are ugly. Infinitely many solutions then means the slopes match and the y-intercepts match too. It is more arithmetic than the ratio method, but it makes the difference from the parallel case impossible to miss, since parallel lines agree on slope and disagree on intercept.

One more form shows up: a system where the equations have already been simplified to something like 0 equals 0 after you eliminate. That is the algebraic signature of infinitely many solutions. If instead the variables vanish and you are left with a false statement such as 0 equals 5, the system has no solution.

The trap

Matching only the x and y coefficients and forgetting that the constant term must scale by the same factor, which turns an infinitely-many answer into a parallel-line system.

Try it

Try one
Real question
10203040506020406080Miles drivenTotal charge (dollars)Company ACompany B(40, 60)

The graph shows the total charge, in dollars, for renting a van for m miles from two companies. What is the total charge, in dollars, at the mileage where the two companies charge the same amount?

A jar holds nickels and dimes in the ratio 3 to 4. The coins are worth $5.50 in all. How many dimes are in the jar?

  • A) 10
  • B) 30
  • C) 40 ✓
  • D) 70

Why: Write the counts as 3k nickels and 4k dimes. Their value is 0.05(3k) + 0.10(4k) = 0.55k = 5.50, so k = 10 and there are 40 dimes. Answering 70 gives the total number of coins. Answering 10 means you reported the ratio multiplier rather than the number of dimes. Answering 30 means you reported the number of nickels instead of dimes. Answering 70 means you reported the total number of coins.

Practice more of these free

Questions students ask

What does it look like algebraically when a system has infinitely many solutions?

If you use elimination and both variables disappear, look at what is left. A true statement such as 0 equals 0 or 6 equals 6 means every point on the line satisfies both equations, so there are infinitely many solutions. A false statement such as 0 equals 9 means no solution.

Which pair of coefficients should I compare first?

Compare the pair that contains no unknown letters, since that gives you the scaling factor for free. Then apply that same factor to every other term to solve for the unknowns. Starting with a pair that contains the unknown forces you into extra algebra.

Guides that cover this

More on systems of linear equations