Parameters & feasible regions · SAT Math

A parameter describes a whole collection of equations. The goal is to decide which coefficient or constant creates the requested behavior. Expanding first reveals whether the variable survives, cancels to an identity, or cancels to a contradiction. A special value may change the number of solutions, so dividing by an expression containing the parameter can lose exactly the case the question is testing.

Match two unknown parameters

For every real x, let 3(x+5)+p(x−2)=7x+c. Expanding the variable terms gives 3+p=7, so p=4. Matching constants then gives c=15−2p=7. Thus p+c=11. Substituting only x=0 would give a relation between p and c without determining both. The identity must match coefficients as well as constants, which distinguishes this task from solving one equation at one input.

A relationship before a coordinate

A perpendicular line to y=2x+1 has slope −1/2. If it passes through (4,7), substitute 7=(−1/2)4+b to get b=9. Reading 7 as the intercept would ignore that the known point has input four. Parallel lines instead use the same slope; decide which geometric condition is actually supplied.

Intersect two feasible boundaries

Let nonnegative integers x,y satisfy 2x+3y≤11 and x+2y≤11, with x=2. The first constraint gives y≤7/3, while the second gives y≤9/2. Both must hold, so the first is tighter and the greatest integer y is two. Choosing four from the second constraint alone gives a point outside the intersection of the allowed regions.

Transfer and check

Change the first constraint to 2x+3y≤20, keeping the second and x=2. The bounds become y≤16/3 and y≤9/2. Now the second constraint is tighter, and the greatest integer y is four. Check both inequalities at y=4 and identify which one fails at y=5. A new budget can change the active boundary without changing the need to satisfy both.

WORKED EXAMPLE

For every real x, 3(x+5)+p(x−2)=7x+c. What is p+c?

Match x coefficients: 3+p=7, so p=4.
Match constants: c=3×5−4×2=7.
The requested sum is p+c=4+7=11.

Answer: 11

Compare the other answer choices

Match the x coefficient but stop at p.

Ignore the constant contribution from the second product.

Use a positive sign for the constant from p(x−2).

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

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