Math  /  Algebra

Questionq=\mathrm{q}= 1148
12. (S pet] The supply aquation and the demand equation for a product are given below. Find the equiltrium quantion Show your work. Solutions by calculator receive no credit. Supply 50039=74500-39=74  Demand 16p+7q=870\text { Demand } 16 p+7 q=870

Studdy Solution

STEP 1

1. The supply equation is given by 32p3q=74 32p - 3q = 74 .
2. The demand equation is given by 16p+7q=870 16p + 7q = 870 .
3. We are tasked with finding the equilibrium quantity, which occurs when supply equals demand.
4. Solutions by calculator are not allowed.

STEP 2

1. Set up the system of equations.
2. Solve the system of equations for p p and q q .
3. Verify the solution to ensure it satisfies both equations.

STEP 3

Set up the system of equations using the given supply and demand equations:
Supply: 32p3q=74 32p - 3q = 74
Demand: 16p+7q=870 16p + 7q = 870

STEP 4

Solve the system of equations for p p and q q .
First, solve one of the equations for one variable. Let's solve the supply equation for p p :
32p=3q+74 32p = 3q + 74
p=3q+7432 p = \frac{3q + 74}{32}

STEP 5

Substitute the expression for p p from the supply equation into the demand equation:
16(3q+7432)+7q=870 16\left(\frac{3q + 74}{32}\right) + 7q = 870
Simplify the equation:
48q+118432+7q=870 \frac{48q + 1184}{32} + 7q = 870
Multiply through by 32 to eliminate the fraction:
48q+1184+224q=27840 48q + 1184 + 224q = 27840
Combine like terms:
272q+1184=27840 272q + 1184 = 27840

STEP 6

Isolate q q by subtracting 1184 from both sides:
272q=278401184 272q = 27840 - 1184
272q=26656 272q = 26656
Divide both sides by 272 to solve for q q :
q=26656272 q = \frac{26656}{272}
q=98 q = 98

STEP 7

Verify the solution by substituting q=98 q = 98 back into the original equations to ensure it satisfies both.
Substitute q=98 q = 98 into the supply equation:
32p3(98)=74 32p - 3(98) = 74
32p294=74 32p - 294 = 74
32p=368 32p = 368
p=36832 p = \frac{368}{32}
p=11.5 p = 11.5
Substitute p=11.5 p = 11.5 and q=98 q = 98 into the demand equation:
16(11.5)+7(98)=870 16(11.5) + 7(98) = 870
184+686=870 184 + 686 = 870
870=870 870 = 870
Both equations are satisfied, confirming the solution.
The equilibrium quantity is:
98 \boxed{98}

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