Problems in Mathematical Terms

Problems in Mathematical Terms exercise

• Identifying independent and dependent variables to write and evaluate equations.

  Hints

  The independent variable is freely chosen and is not determined by any other variable.

  The dependent variable changes every time there is a change to the independent variable.

  For example: If the color of a Golem is 4 times the number of pumpkins, the equation can be represented by $c=4p$, where $c$ is the color of the Golem and $p$ is the number of pumpkins.

  Solution

  Independent variables are chosen freely and can change without the influence of anything else in the equation. The examples of independent variables used here are:

  • eyes of newt
  • bloodroots

  Dependent variables change every time there is a change with the independent variable. This means they depend on the independent variable. The examples of dependent variables used here are:

  • size of the Golem
  • strength of the Golem

  Magic Spell #1: For every eyes of newt Wanda has, the Golem will be 3 feet tall.

  • The equation is $h=3n$
  • If $n=1$ then $h=3$
  • If $n=4$ then $h=12$
  • Using the equation $h=3n$ we substitute $12$ for height and recognize that means Wanda needs $4$ eyes of newt for her potion.

  Magic Spell #2: For every bloodroot Wanda adds to the potion, the Golem will gain the strength of 5 men.

  • The equation is $s=5b$.
  • If $b=1$ then $s=5$.
  • If $b=5$ then $s=25$.
  • Use the equation $s=5b$ and substitute $5$ for bloodroots to give Wanda's Golem the strength of $25$ men.

• Decide which statements about variables and constants are true.

  Hints

  In the equation, $h=3n$, $n$ is an independent variable because it can be freely chosen.

  In the equation, $ y=x+1$, $1$ is a constant.

  In the equation $h=3n$, $h$ is the dependent variable because it depends on the independent variable, $n$.

  Solution

  A constant is a value that stays the same and does not change.

  Independent variables can be freely chosen and do not depend on any other variable.

  Dependent variables are determined by independent variables.

  Wanda's magic book states:

  • The eyes, fingers and hat of a Golem never change.
  • A Golem's size is based on the number of eyes of newt.
  • A Golem's strength is based on the number of bloodroots.

  The following statements are true:

  • The size and strength of a Golem are dependent variables because size and strength are determined on the number of eyes of newt and bloodroots that Wanda chooses.
  • The eyes, fingers and hat on a Golem are constants because they never change.
  • Bloodroots and eyes of newt are independent variables because they are freely chosen.
  • There are two types of variables: independent and dependent.

  The following statements are false:

  • Independent variables are determined by dependent variables is false because dependent variables are determined by independent variables.
  • The size of a golem is an independent variable is false because the size of a Golem is a dependent variable.
  • Constants can change is false because constants never change.

• Identify the dependent and independent variables.

  Hints

  The independent variable is freely chosen and is not determined by any other variable.

  The dependent variable is determined by the independent variable.

  In the equation $k=2t$, $t$ is the variable that is freely chosen, and $k$ is the variable that depends on $t$.

  Solution

  1. $d=2t$

  • $d$ is the dependent variable because the donation amount is determined by how many tickets they sell. $t$ is the independent variable because Ella and Jahmal can choose how many tickets they want to sell.

  2. $c=3.50b$

  • $c$ is the dependent variable because the cost depends on how many boxes, $b$, the independent variable, Antonio buys.

  3. $f=4p$

  • $f$ is the is the dependent variable because the number of flowers depends on how many pots, $p$, the independent variable, Maya buys.

• Find the parts of the story which correspond to different mathematical terms.

  Hints

  Mi Xuan is a photographer. She charges a one time sitting fee of 30 dollars. Then it costs 15 dollars per photo that you buy. The total cost for photos by Mi Xuan can be found by creating an equation.

  • The independent variable in this situation is the the number of photos you choose. This is represented by per photo in the description.

  Mi Xuan is a photographer. She charges a one time sitting fee of 30 dollars. Then it costs 15 dollars per photo that you buy. The total cost for photos by Mi Xuan can be found by creating an equation.

  • The dependent variable in this situation is the the total cost because it depends how many photos you choose to buy.

  Mi Xuan is a photographer. She charges a one time sitting fee of 30 dollars. Then it costs 15 dollars per photo that you buy. The total cost for photos by Mi Xuan can be found by creating an equation.

  • The constant term in this situation is the the one time sitting fee of 30 dollars because it doesn't change.

  Solution

  1. Kai is doing a science experiment for school. She starts with water that is $75^{\circ}$F. Every time she heats the water in the microwave the overall temperature increases 10 degrees per minute.

  • per minute is the independent variable because Kai can choose how many minutes she wants to heat the water in the microwave.
  • overall temperature is the dependent variable because the temperature of the water is determined by how many minutes Kai heats the water.
  • 75 is the constant term because the original temperature of the water never changes.

  2. Jackson wants to purchase a gym membership. He doesn't know the total cost of the membership. However, he does know there is a one time $10$ dollar registration fee, and a monthly fee of $25$ dollars.

  • monthly fee is the independent variable because Jackson can choose the number of months he would like a gym membership.
  • total cost of the membership is the dependent variable because the membership cost is determined by how many months Jackson uses the gym.
  • 10 is the constant term because it is a one-time membership fee that never changes.

• Determine which of the following are independent or dependent variables, or constant terms.

  Hints

  In mathematics, a constant term is a term in an algebraic expression that has a value that is constant or cannot change. This is because it stands alone, and does not contain any variables.

  Independent variables are freely chosen and do not depend on any other variable.

  Dependent variables are determined by the independent variables.

  Solution

  1. Constant Term

  • $3$ in $y=2x+3$
  • $3$ is the only term that never changes in the equation and is therefore called a constant term. No matter what the values $x$ and $y$ are, 3 will never change.
  • The $2$, in $y=2x+3$, is the coefficient of $x$.

  2. Dependent Variable

  • The $y$ in $y=6x$.
  • The $y$ is the dependent variable because it changes based on $x$.
  • For example, if we choose $x$ to be 2, then $y=6(2)$ which is $12$.
  • If we choose $x$ to be $-3$, then $y=6(-3)$ which is $-18$.
  • Therefore, $y$ is dependent on the independent variable, $x$.

  3. Independent Variable

  • The $x$ in $y=7x+1$.
  • The $x$ is called the independent variable because we can freely choose its value and it is not determined by any other variable.
  • For example, we can choose $x$ to be $1$, $\frac{1}{2}$, $-2$, and so on, to find as many values as we want for $y$.

  4. Variables

  • The $x$ and $y$ in $y=-2x+9$.
  • The $x$ and $y$ are called variables because they can change or vary.

• Set up the linear equation.

  Hints

  • A constant term is a number that never changes.
  • The independent variable is chosen freely.
  • The dependent variable is determined by the independent variable.

  The cost of $\$ 3$ per shirt plus a $\$ 2$ donation is represented by the equation $c=3s+2$, where $c$ is the cost and $s$ is the number of shirts.

  The cost of $\$ 0.99$ per mango plus a $\$ 2$ discount is represented by the equation $c=0.99m-2$ or $c=0.99m+-2$, where $c$ is the cost and $m$ is the number of mangos.

  Solution

  1. $c=2d+175$

  • The dependent variable is $c$ because it depends on $d$.
  • The independent variable is $d$ because it can be freely chosen.
  • The coefficient of the independent variable is $2$.
  • The constant term is $175$ because it never changes.

  2. $c=1.25t+10$

  • The dependent variable is $c$ because it depends on $t$.
  • The independent variable is $t$ because it can be freely chosen.
  • The coefficient of the independent variable is $1.25$.
  • The constant term is $10$ because it never changes.

  3. $c=1.50p+-5$

  • The dependent variable is $c$ because it depends on $p$.
  • The independent variable is $p$ because it can be freely chosen.
  • The coefficient of the independent variable is $1.50$.
  • The constant term is $-5$ because it never changes.