SAT · Algebra · TOPIC LESSON
Combine two constraints and understand whether they meet once, never or everywhere.
BY THE END, YOU CAN
BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This lesson supports SAT Math and Reading and Writing skill study; it does not reproduce official questions or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
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01 / UNDERSTAND & APPLY
A system combines equations that must hold at the same time. Its solution is an ordered pair, not one answer from each equation separately. Geometrically, common points lie on both lines. One intersection gives one solution; distinct parallel lines give none; coincident lines give infinitely many.
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02 / UNDERSTAND & APPLY
If one equation isolates a variable, substitute that expression into the other equation. Use parentheses to preserve subtraction and multiplication. Solve the resulting one-variable equation, recover the other variable, and verify the pair in both original equations.
ORIGINAL WORKED EXAMPLE
Solve y = 2x − 1 and 3x + y = 19.
Answer (x,y) = (4,7)
03 / UNDERSTAND & APPLY
Adding equal quantities on both sides of equations produces another true equation. Multiply an entire equation by a nonzero constant if necessary, then add or subtract to cancel one variable. Multiplying only a variable term changes the original condition.
ORIGINAL WORKED EXAMPLE
Solve 2x + 3y = 17 and 5x + 6y = 37.
Answer (3,11/3)
04 / UNDERSTAND & APPLY
Define variables and units. A total count and total cost often give different equations: the count equation has coefficients 1, while the cost equation uses unit prices. Independence matters: repeating a condition in scaled form adds no new information.
ORIGINAL WORKED EXAMPLE
A club buys 14 notebooks. Basic notebooks cost $3 and large notebooks cost $5; the total is $54. How many of each?
Answer 8 basic and 6 large notebooks
05 / UNDERSTAND & APPLY
After elimination, 0 = a nonzero number is a contradiction and means no solution. The statement 0 = 0 means one equation repeated the other; it leaves infinitely many pairs on the common line. It does not mean the only pair is (0,0).
For ax + by = c and dx + ey = f, the determinant ae − bd being nonzero guarantees one solution. If it is zero, compare the constants too; zero determinant alone does not distinguish no solution from infinitely many.
06 / UNDERSTAND & APPLY
To make two lines parallel, make their variable coefficients proportional. To make them coincide, the constants must have the same proportionality. Check exceptional zero coefficients directly rather than dividing by a parameter that might be zero.
ORIGINAL WORKED EXAMPLE
For which k does 2x + 3y = 7 and 4x + ky = 15 have no solution?
Answer k = 6
07 / UNDERSTAND & APPLY
If the question asks for x + y or another expression, add or subtract the original equations to form it directly. This is still valid elimination; it avoids extra work but must produce exactly the requested combination.
ORIGINAL WORKED EXAMPLE
If 4x + 3y = 26 and 2x + 3y = 16, find x + y.
Answer 7
08 / UNDERSTAND & APPLY
A graphing calculator can locate line intersections and check an algebraic pair. Distinct nearly parallel lines can meet far outside the default window; not seeing a crossing is not proof that none exists. Overlapping screen traces suggest coincidence but exact coefficient comparison is stronger.
In context, reject impossible counts, but do not quietly round a fractional solution into an integer. It may reveal that the stated assumptions do not admit a physical solution. Keep exact fractions through the algebra.
In the notebook example, the count and cost constraints are lines in the (b,l) plane. Their intersection (8,6) means both the total count and the budget match. An intersection outside the nonnegative quadrant would solve the equations but would not describe a possible purchase.
09 / UNDERSTAND & APPLY
For every solution pair, substitute into both original equations.
ORIGINAL PRACTICE · 1
(6,3)
Adding gives 2x = 12, hence x = 6; then y = 3.
ORIGINAL PRACTICE · 2
(3,9)
Substitution yields 4x = 12, so x = 3 and y = 9.
ORIGINAL PRACTICE · 3
None
Twice the first left side must equal 10, not 11.
ORIGINAL PRACTICE · 4
Infinitely many
The second equation is twice the entire first equation, so they represent one line.
ORIGINAL PRACTICE · 5
c = 18
Every coefficient and constant in the second equation must be three times the first.
ORIGINAL PRACTICE · 6
6 adult tickets
Let a + c = 10 and 10a + 4c = 76. Substitute c = 10 − a: 6a + 40 = 76, so a = 6.
ORIGINAL PRACTICE · 7
3
Adding eliminates y and gives 8x = 24.
ORIGINAL PRACTICE · 8
No
An intersection may lie outside the window; compare slopes or eliminate exactly to classify the system.
KEEP THE BIG IDEAS
READ WITH CONTEXT
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