PROBLEM_1. Tickets cost $A1.toFixed( 2 ) each for adults and $B1.toFixed( 2 ) each for kids, and the
group paid $C1.toFixed( 2 ) in total. There were abs( C2 ) fewer adults than kids in the group.
PROBLEM_2.C1 people attended
a baseball game. Everyone there was a fan of either the home team or the away team. The number of home team fans was abs( C2 ) less than abs( B2 ) times the number of away team fans.
How
many home team and away team fans attended the game?
# of UNIT_1 = X
# of UNIT_2 = Y
Let x equal the number of UNIT_1 and y equal the number of UNIT_2.
The system of equations is then:
\color{BLUE}{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}
x = expr(["+", ["*", -B2, "y"], C2])
Solve for x and y using substitution.
Since x has already been solved for, substitute expr(["+", ["*", -B2, "y"], C2]) for x in the first equation.
\color{BLUE}{A1-}\color{GREEN}{(expr(["+", ["*", -B2, "y"], C2]))}\color{BLUE}{+ expr(["*", B1, "y"]) = C1}
Simplify and solve for y.
expr(["+", ["*", roundTo( 8, A1 * -B2 ), "y"], roundTo( 8, A1 * C2 )]) + expr(["*", B1, "y"]) = C1
expr(["+", ["*", roundTo( 8, A1 * -B2 + B1 ), "y"], roundTo( 8, A1 * C2 )]) = C1
expr(["+", ["*", roundTo( 8, A1 * -B2 + B1 ), "y"], A1 * C2])\color{BLUE}{SIGN_1abs( roundTo( 8, A1 * C2 ) )} = C1\color{BLUE}{SIGN_1abs( roundTo( 8, A1 * C2 ) )}
expr(["*", roundTo( 8, A1 * -B2 + B1 ), "y"]) = roundTo( 8, C1 - A1 * C2 )
\dfrac{expr(["*", roundTo( 8, A1 * -B2 + B1 ), "y"])}{\color{BLUE}{roundTo( 8, A1 * -B2 + B1 )}} = \dfrac{roundTo( 8, C1 - A1 * C2 )}{\color{BLUE}{roundTo( 8, A1 * -B2 + B1 )}}
\color{ORANGE}{y = Y}
Now that you know \color{ORANGE}{y = Y}, plug it back into \thinspace \color{GREEN}{x = expr(["+", ["*", -B2, "y"], C2])}\thinspace to find x.
\color{GREEN}{x = -B2-}\color{ORANGE}{(Y)}\color{GREEN}{ + C2}
x = -B2 * Y + C2
\color{red}{x = X}
You can also plug \color{ORANGE}{y = Y} into \thinspace \color{BLUE}{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}\thinspace and get the same answer for x:
\color{BLUE}{expr(["*", A1, "x"]) + B1-}\color{ORANGE}{(Y)}\color{BLUE}{= C1}
\color{red}{x = X}
There were X UNIT_1 and Y UNIT_2.
PROBLEM_1. Bags of candy cost $A1.toFixed( 2 ), and bags of cookies cost $B1.toFixed( 2 ), and sales equaled $C1.toFixed( 2 ) in total. There were C2 more bags of cookies than candy sold.
PROBLEM_2.The sum of two angles' measures is C1 degrees. Angle 2 is abs( C2 ) degrees smaller than abs( A2 ) times angle 1.
What are the measures of the two angles in degrees?
# of UNIT_1 = X
# of UNIT_2 = Y
Let x equal the numbermeasure of UNIT_1 and y equal the numbermeasure of UNIT_2.
The system of equations is then:
\color{BLUE}{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}
y = expr(["+", ["*", -A2, "x"], C2])
Since we already have solved for y in terms of x, we can use substitution to solve for x and y.
Substitute expr(["+", ["*", -A2, "x"], C2]) for y in the first equation.
\color{BLUE}{expr(["*", A1, "x"]) + B1-}\color{GREEN}{(expr(["+", ["*", -A2, "x"], C2]))}\color{BLUE}{= C1}
Simplify and solve for x.
expr(["+", ["*", A1, "x"], ["*", roundTo( 8, B1 * -A2 ), "x"]]) + roundTo( 8, B1 * C2 ) = C1
expr(["+", ["*", roundTo( 8, A1 + B1 * -A2 ), "x"], roundTo( 8, B1 * C2 )]) = C1
expr(["+", ["*", roundTo( 8, A1 + B1 * -A2 ), "x"], roundTo( 8, B1 * C2 )])\color{BLUE}{SIGN_1abs( roundTo( 8, B1 * C2 ) )} = C1\color{BLUE}{SIGN_1abs( roundTo( 8, B1 * C2 ) )}
expr(["*", roundTo( 8, A1 + B1 * -A2 ), "x"]) = roundTo( 8, C1 - B1 * C2 )
\dfrac{expr(["*", roundTo( 8, A1 + B1 * -A2 ), "x"])}{\color{BLUE}{roundTo( 8, A1 + B1 * -A2 )}} = \dfrac{roundTo( 8, C1 - B1 * C2 )}{\color{BLUE}{roundTo( 8, A1 + B1 * -A2 )}}
\color{red}{x = X}
Now that you know \color{red}{x = X}, plug it back into \thinspace \color{GREEN}{y = expr(["+", ["*", -A2, "x"], C2])}\thinspace to find y.
\color{GREEN}{y = -A2-}\color{red}{(X)}\color{GREEN}{ + C2}
y = roundTo( 8, -A2 * X ) + C2
y = Y
You can also plug \color{red}{x = X} into \thinspace \color{BLUE}{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}\thinspace and get the same answer for y:
\color{BLUE}{A1-}\color{red}{(X)}\color{BLUE}{ + expr(["*", B1, "y"]) = C1}
\color{ORANGE}{y = Y}
X bags of candy and Y bags of cookies were sold.The measure of angle 1 is X^{\circ} and the measure of angle 2 is Y^{\circ}.
All of the {3rd|4th|5th} grade teachers and students from school(1) went on a field trip to an {art|archaeology} museum. Tickets were $A1.toFixed( 2 ) each for teachers and $B1.toFixed( 2 ) each for students, and the
group paid $C1.toFixed( 2 ) in total.
{A few weeks later|The next month}, the same group visited a {science|natural history} museum where the tickets cost $A2.toFixed( 2 ) each for teachers and $B2.toFixed( 2 ) each for students, and the
group paid $C2.toFixed( 2 ) in total.
Find the number of teachers and students on the field trips.
# of teachers = X
# of students = Y
Let x equal the number of teachers and y equal the number of students.
The system of equations is:
\color{BLUE}{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}
expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) = C2
Solve for x and y using elimination.
Multiply the bottomtop equation by MULT2MULT1 and the bottom equation by MULT2.
\color{BLUE}{expr(["+", ["*", A1 * MULT1, "x"], ["*", B1 * MULT1, "y"]]) = C1 * MULT1}
expr(["+", ["*", A2 * MULT2, "x"], ["*", B2 * MULT2, "y"]]) = C2 * MULT2
Add the top and bottom equations together.
expr(["*", roundTo( 8, B1 * MULT1 + B2 * MULT2 ), "y"]) = roundTo( 8, C1 * MULT1 + C2 * MULT2 )
\dfrac{expr(["*", roundTo( 8, B1 * MULT1 + B2 * MULT2 ), "y"])}{\color{BLUE}{roundTo( 8, B1 * MULT1 + B2 * MULT2 )}} = \dfrac{roundTo( 8, C1 * MULT1 + C2 * MULT2 )}{\color{BLUE}{roundTo( 8, B1 * MULT1 + B2 * MULT2 )}}
\color{ORANGE}{y = Y}
Now that you know \color{ORANGE}{y = Y}, plug it back into \thinspace \color{BLUE}{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}\thinspace to find x.
\color{BLUE}{expr(["*", A1, "x"]) + B1-}\color{ORANGE}{(Y)}\color{BLUE}{= C1}
expr(["+", ["*", A1, "x"], B1 * Y]) = C1
expr(["+", ["*", A1, "x"], B1 * Y])\color{BLUE}{SIGN_1abs( B1 * Y )} = C1\color{BLUE}{SIGN_1abs( B1 * Y )}
expr(["*", A1, "x"]) = roundTo( 8, C1 - B1 * Y )
\dfrac{expr(["*", A1, "x"])}{\color{BLUE}{A1}} = \dfrac{roundTo( 8, C1 - B1 * Y )}{\color{BLUE}{A1}}
\color{red}{x = X}
You can also plug \color{ORANGE}{y = Y} into \thinspace \color{GREEN}{expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) = C2}\thinspace and get the same answer for x:
\color{GREEN}{expr(["*", A2, "x"]) + B2-}\color{ORANGE}{(Y)}\color{GREEN}{= C2}
\color{red}{x = X}
There were X teachers and Y students on the field trips.
The sum of two numbers is C1, and their difference is C2. What are the two numbers?
x =
y =
Let x be the first number, and let y be the second number.
The system of equations is:
\color{BLUE}{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}
expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) = C2
Solve for x and y using elimination.
Multiply the bottomtop equation by MULT2MULT1 and the bottom equation by MULT2.
\color{BLUE}{expr(["+", ["*", A1 * MULT1, "x"], ["*", B1 * MULT1, "y"]]) = C1 * MULT1}
expr(["+", ["*", A2 * MULT2, "x"], ["*", B2 * MULT2, "y"]]) = C2 * MULT2
Add the top and bottom equations together.
expr(["*", roundTo( 8, A1 * MULT1 + A2 * MULT2 ), "x"]) = roundTo( 8, C1 * MULT1 + C2 * MULT2 )
\dfrac{expr(["*", roundTo( 8, A1 * MULT1 + A2 * MULT2 ), "x"])}{\color{BLUE}{roundTo( 8, A1 * MULT1 + A2 * MULT2 )}} = \dfrac{roundTo( 8, C1 * MULT1 + C2 * MULT2 )}{\color{BLUE}{roundTo( 8, A1 * MULT1 + A2 * MULT2 )}}
\color{red}{x = X}
Now that you know \color{red}{x = X}, plug it back into \thinspace \color{BLUE}{expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) = C1}\thinspace to find y.
\color{BLUE}{A1-}\color{red}{(X)}\color{BLUE}{ + expr(["*", B1, "y"]) = C1}
expr(["+", A1 * X, ["*", B1, "y"]]) = C1
A1 * X\color{BLUE}{SIGN_1abs( A1 * X )} + expr(["*", B1, "y"]) = C1\color{BLUE}{SIGN_1abs( A1 * X )}
expr(["*", B1, "y"]) = roundTo( 8, C1 - A1 * X )
\dfrac{expr(["*", B1, "y"])}{\color{BLUE}{B1}} = \dfrac{roundTo( 8, C1 - A1 * X )}{\color{BLUE}{B1}}
\color{ORANGE}{y = Y}
You can also plug \color{red}{x = X} into \thinspace \color{GREEN}{expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) = C2}\thinspace and get the same answer for y:
\color{GREEN}{A2-}\color{red}{(X)}\color{GREEN}{ + expr(["*", B2, "y"]) = C2}
\color{ORANGE}{y = Y}