Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Dr. Patrick, a pediatrician, has $3$ annual checkups and $2$ sick visits scheduled next Tuesday, which will fill a total of $176$ minutes on her schedule. Next Wednesday, she has $2$ annual checkups and $1$ sick visit on the schedule, which should take $112$ minutes. How much time is allotted for each type of appointment? $\newline$ The time allotted is $\_$ minutes for an annual checkup and $\_$ minutes for a sick visit.

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Algebra 1

Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Dr. Patrick, a pediatrician, has

3

annual checkups and

2

sick visits scheduled next Tuesday, which will fill a total of

176

minutes on her schedule. Next Wednesday, she has

2

annual checkups and

1

sick visit on the schedule, which should take

112

minutes. How much time is allotted for each type of appointment?

\newline

The time allotted is

\_

minutes for an annual checkup and

\_

minutes for a sick visit.

Define variables: Define the variables for the types of appointments. $\newline$ Let $x$ be the time allotted for an annual checkup. $\newline$ Let $y$ be the time allotted for a sick visit.
Write equations: Write the system of equations based on the given information. $\newline$ For Tuesday: $3$ annual checkups and $2$ sick visits take $176$ minutes. $\newline$ For Wednesday: $2$ annual checkups and $1$ sick visit take $112$ minutes. $\newline$ This gives us the system: $\newline$ $3x + 2y = 176$ $\newline$ $2x + y = 112$
Multiply and align coefficients: Multiply the second equation by $2$ to align the coefficients of $y$ for elimination. $\newline$ $2(2x + y) = 2(112)$ $\newline$ $4x + 2y = 224$
Eliminate variable: Subtract the second equation from the first equation to eliminate $y$ . $\newline$ $(3x + 2y) - (4x + 2y) = 176 - 224$ $\newline$ $3x - 4x + 2y - 2y = -48$ $\newline$ $-x = -48$
Solve for x: Solve for x. $\newline$ $-x = -48$ $\newline$ Multiply both sides by $-1$ to get the value of x. $\newline$ $x = 48$
Substitute $x$ into equation: Substitute the value of $x$ into the second original equation to solve for $y$ . $\newline$ $2x + y = 112$ $\newline$ $2(48) + y = 112$ $\newline$ $96 + y = 112$
Solve for y: Subtract $96$ from both sides to solve for $y$ . $\newline$ $96 + y - 96 = 112 - 96$ $\newline$ $y = 16$
Verify solution: Verify the solution by substituting $x$ and $y$ into both original equations. $\newline$ First equation: $3x + 2y = 176$ $\newline$ $3(48) + 2(16) = 176$ $\newline$ $144 + 32 = 176$ $\newline$ $176 = 176$ $\newline$ Second equation: $2x + y = 112$ $\newline$ $2(48) + 16 = 112$ $\newline$ $96 + 16 = 112$ $\newline$ $112 = 112$ $\newline$ Both equations are true, so the solution is correct.