Question 297·Hard·Algebra

0 / 700

The values of two linear functions $f$ and $g$ are shown below.

x	3	11
$f(x)$	$-7$	$17$

x	2	8
$g(x)$	$20$	$5$

For what value of $x$ does $f(x)=g(x)$ ?

Two rows for each linear function give enough information to find its slope, the output change per unit of input. Build each rule from its own rows, then set the outputs equal at the same input. Desmos can solve that equality and show the result as an exact fraction. Mixing rows from the two tables gives a slope that belongs to neither function.

Hints

Hint 1
A slope is the change in output divided by the change in input. For $f$ , subtract its first output from its second, then subtract the matching inputs in the same order. What goes above and below the fraction bar?
Hint 2
Keep the two rows for $g$ together. Its output falls while its input rises, so its slope is negative. Use the output change $5-20$ over the input change $8-2$ .
Hint 3
A point-slope rule uses a known row and the slope: $f(x)=m(x-x_0)+f(x_0)$ . Build one rule for each function, then find the shared input where their outputs are equal.

Step-by-step

Build the two rules and solve in Desmos
Step 1
Find the slope of f
The slope is output change divided by input change. For the $f$ table, type $\frac{17-(-7)}{11-3}$ in Desmos. It prints $3$ , so $f$ rises by $3$ whenever its input rises by $1$ . Subtract the inputs in the same order as the outputs.
Step 2
Find the slope of g
Use only the $g$ rows this time. Type $\frac{5-20}{8-2}$ ; Desmos gives $-2.5$ , or $-\frac52$ . The negative slope fits the table: $g$ falls as its input rises.
Step 3
Write the rule for f
The $f$ table says $f(3)=-7$ . Starting there and changing the output by $3$ per input unit gives the point-slope rule $f(x)=3(x-3)-7$ . Type it in Desmos. The $x-3$ makes the slope term zero at the known input $3$ , leaving the known output $-7$ .
Step 4
Write the rule for g
The $g$ table gives $g(2)=20$ , and its slope is $-2.5$ . Type $g(x)=-2.5(x-2)+20$ . This point-slope rule gives the known output $20$ when $x=2$ , then falls by $2.5$ per input unit.
Step 5
Find the shared input
Type $f(x_1)\sim g(x_1)$ to find where the outputs are equal. Use $x_1$ because Desmos graphs with $x$ ; the $\sim$ tells it to fit an equality. Under PARAMETERS, Desmos gives $x_1\approx7.454545\ldots$ . Don't round that to $7.5$ : the choices ask for an exact value.
Step 6
Read the exact value
Type $x_1$ on its own line and tap its fraction button. Desmos shows $\frac{82}{11}$ , the input where the two functions have the same output. Choice B.

Question 297·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this