Question 52·SAT Desmos Regression Practice·Algebra

Infinitely many solutions means the equation works for every $x$. Both sides are linear expressions: $x$ appears only to the first power. Two linear expressions that agree at two different inputs must be identical, so testing several inputs can determine both constants.

Type $x_1=[1...5]$. Desmos displays $[1,2,3,4,5]$, giving the regression five different inputs. A single input would leave two unknown constants to find.

Type $a(4-2x_1)+b(x_1+1)\sim7-x_1$. The $\sim$ tells Desmos to fit $a$ and $b$ across the input list. Under PARAMETERS, it shows $a\approx1.3333$ and $b\approx1.6667$. Those are rounded values, so check their fraction forms before choosing.

Type $a$ and $b$ on their own lines and tap each line’s fraction button. Desmos shows $a=\frac43$ and $b=\frac53$. These are the values of the constants that make the equation hold for every $x$. Choice D.

Two linear expressions that agree at two distinct inputs agree everywhere: their difference is linear, and a nonzero linear expression can equal zero at only one input. Choose $x=0$ first because it removes the $x$-terms. Substitute it:

Simplify:

Choose the distinct input $x=1$ to get another equation. Substitute it:

Simplify:

Divide every term of $2a+2b=6$ by $2$:

Subtract $a+b=3$ from $4a+b=7$. The $b$-terms cancel, leaving one unknown:

Simplify:

Divide both sides by $3$:

Use $a+b=3$ because it has fewer terms. Substitute $a=\frac43$:

Subtract $\frac43$ from both sides:

Write $3$ in thirds and subtract:

So $a=\frac43$ and $b=\frac53$ make both sides identical. Choice D.