$3$ . Regents Question. $\newline$ Two friends went to a restaurant and ordered one plain pizza and two sodas. Their bill totaled $\$ 15.95$ . Later that day, five friends went to the same restaurant. They ordered three plain pizzas and each person had one soda. Their bill totaled $\$ 45.90$ . $\newline$ Write and solve a system of equations to determine the price of one plain pizza. [Only an algebraic solution can receive full credit.]

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3

. Regents Question.

\newline

Two friends went to a restaurant and ordered one plain pizza and two sodas. Their bill totaled

\$ 15.95

. Later that day, five friends went to the same restaurant. They ordered three plain pizzas and each person had one soda. Their bill totaled

\$ 45.90

\newline

Write and solve a system of equations to determine the price of one plain pizza. [Only an algebraic solution can receive full credit.]

Forming Equations: Let's denote the price of one plain pizza as $P$ and the price of one soda as $S$ . We can create two equations based on the information given. $\newline$ The first scenario with two friends: $1P + 2S = (\$)15.95$ $\newline$ The second scenario with five friends: $3P + 5S = (\$)45.90$ $\newline$ We will use these two equations to form a system of equations.
Solving for S: We can start by solving one of the equations for one of the variables. Let's solve the first equation for S. $\newline$ $1P + 2S = \$15.95$ $\newline$ $2S = \$15.95 - 1P$ $\newline$ $S = (\$15.95 - 1P) / 2$ $\newline$ Now we have an expression for S in terms of P.
Substitute and Simplify: Next, we substitute the expression for $S$ into the second equation to solve for $P$ . $3P + 5S = \$(45.90)$ Substitute $S$ from the first equation: $3P + 5\left(\frac{\$(15.95) - 1P}{2}\right) = \$(45.90)$ Now we need to simplify and solve for $P$ .
Solving for P: Let's distribute the $5$ inside the parentheses: $\newline$ $3P + (5 \times \$(15.95))/2 - (5 \times 1P)/2 = \$(45.90)$ $\newline$ Now we simplify the equation further. $\newline$ $3P + \$(39.875) - 2.5P = \$(45.90)$ $\newline$ Combine like terms: $\newline$ $0.5P + \$(39.875) = \$(45.90)$
Final Check: Now we solve for $P$ by subtracting $\$39.875$ from both sides of the equation: $\newline$ $0.5P = \$45.90 - \$39.875$ $\newline$ $0.5P = \$6.025$ $\newline$ To find $P$ , we divide both sides by $0.5$ : $\newline$ $P = \$6.025 / 0.5$ $\newline$ $P = \$12.05$
Final Check: Now we solve for $P$ by subtracting $\$39.875$ from both sides of the equation: $\newline$ $0.5P = \$45.90 - \$39.875$ $\newline$ $0.5P = \$6.025$ $\newline$ To find $P$ , we divide both sides by $0.5$ : $\newline$ $P = \$6.025 / 0.5$ $\newline$ $P = \$12.05$ We have found the price of one plain pizza, which is $\$12.05$ . To ensure there are no math errors, we can plug this value back into one of the original equations to see if it makes sense. $\newline$ Let's use the first equation: $\newline$ $1P + 2S = \$15.95$ $\newline$ $1(\$12.05) + 2S = \$15.95$ $\newline$ $\$12.05 + 2S = \$15.95$ $\newline$ Now we solve for $S$ : $\newline$ $2S = \$15.95 - \$12.05$ $\newline$ $2S = \$3.90$ $\newline$ $S = \$3.90 / 2$ $\newline$ $0.5P = \$6.025$ $0$ $\newline$ This value for $S$ makes sense with the given information, so there are no math errors.