Question 1056·Hard·Math

0 / 2000

For real constants $a$ and $b$ , the following system of equations has the solution $(3,1)$ .

\begin{aligned} ax+by &= 10 \\[1.4em] bx+ay &= 14 \end{aligned}

If $(p,q)$ is the solution to the following system, what is the value of $p$ ?

\begin{aligned} ap+bq &= 26 \\[1.4em] bp-aq &= 14 \end{aligned}

Your answer

An ordered-pair solution makes both equations true, so a supplied pair can determine constants reused in another system. Substitute the pair, then use a Desmos list regression to find the constants and a second one to find the requested coordinate. Keep the minus sign in the new system; changing it gives a different value.

Hints

Hint 1
An ordered-pair solution tells you which number belongs to each variable. Put $x=3$ and $y=1$ into both equations of the first system. What two equations does that give for $a$ and $b$ ?
Hint 2
A Desmos list regression can find $a$ and $b$ together. Put the two left sides in one list and their matching right sides in another. Desmos matches the entries in order.
Hint 3
Before solving the second system, replace both constants with the values you found. Keep the subtraction in $bp-aq=14$ . Then put the two new left sides in a list and match them to $[26,14]$ .

Step-by-step

Approach 1: Fit the constants, then fit the new solution

Step 1
Use the given pair to make equations for the constants
A solution makes both equations true. Put $x=3$ and $y=1$ into the first system:
$\begin{aligned} 3a+b&=10\\[1.4em] 3b+a&=14. \end{aligned}$
The given pair lets you find the shared constants before solving the new system.
Step 2
Find both constants in Desmos
Type $[3a+b,3b+a]\sim[10,14]$ . The list regression matches the left sides to $10$ and $14$ in order; $\sim$ tells Desmos to find values that fit. Under PARAMETERS, it gives $a=2$ and $b=4$ .
Step 3
Put the constants into the second system
Replace $a$ with $2$ and $b$ with $4$ in both equations:
$\begin{aligned} 2p+4q&=26\\[1.4em] 4p-2q&=14. \end{aligned}$
Keep the minus before $aq$ ; changing it would change the solution.
Step 4
Read the value of p
Type $[2p_1+4q_1,4p_1-2q_1]\sim[26,14]$ . The subscripts let Desmos fit $p_1$ and $q_1$ as unknown values. Under PARAMETERS, it gives $p_1=5.4$ and $q_1=3.8$ . The first fitted value is $p$ , so $p=5.4$ . Grid in 5.4 (or 27/5).

Approach 2: Eliminate q for an exact fraction

Step 1
Make the q-terms opposites
You can use elimination on the second system instead. Its equations are $2p+4q=26$ and $4p-2q=14$ . Multiply every term of the second equation by $2$ so its $q$ -term will cancel the $+4q$ :
$8p-4q=28.$
Step 2
Add to remove q
Add the new equation to $2p+4q=26$ . The opposite $q$ -terms cancel:
$(2p+4q)+(8p-4q)=26+28$ $10p=54.$
Step 3
Divide to isolate p
Divide both sides by $10$ to get the requested coordinate:
$p=\frac{54}{10}.$
Step 4
Give p in simplest form
Divide the numerator and denominator by $2$ :
$p=\frac{27}{5}.$
This is the first coordinate of the second system’s solution. Grid in 27/5 (or 5.4).

Question 1056·Hard·Math

Strategy

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Step-by-step

Lessons that teach this

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