Standard Form

Standard Form exercise

• Determine the number of glasses of lemonade that are sold.

  Hints

  The standard form:

  $Ax+By=C$

  $A\neq 0$ and $B\neq 0$.

  In this example, $A=2$ and $B=1.5$ are the known prices.

  To solve an equation, first simplify by combining like terms, and then isolate the variable using opposite operations.

  Solution

  To write the in standard form:$Ax+By=C$, consider the facts.

  • $x$ is the unknown number of sold glasses of lemonade.
  • $y$ is the number of sold cookies.
  • $A=2$ is the price of one glass of lemonade.
  • $B=1.5$ is the price of one cookie.
  • $C=180$ is the total amount Kayla earned that day.

  The equation written in standard form:

  $2x+1.5y=180$.

  Since we already know the number of sold cookies ($80$), we can substitute $y$ with this number:

  $2x+1.5\times 80=180$.

  Now we can solve the equation:

  $\begin{array}{rclrl} 2x+1.5\times 80&=&~~~180&&|\text{ multiplying}\\ 2x +120&=&~~~180\\ \color{#669900}{-120}&&\color{#669900}{-120}\\ 2x&=&~~~~~60 \end{array}$

  Finally we divide both sides of the equation by $2$:

  $x=30$.

  Now we know that Kayla sold 30 glasses of lemonade.

• Describe how to graph the equation.

  Hints

  Multiplying by $0$ always equals $0$.

  To solve the equation $3x=12$, you have to divide by $3$.

  Each point $P(p_x,p_y)$ has an x-($p_x$) and a y-coordinate ($p_y$).

  Solution

  To graph the equation $2x+1.5y=180$, let's evaluate the intercepts:

  • for the y-intercept substitute $0$ for $x$
  • for the x-intercept substitute $0$ for $y$

  y-intercept

  • $2(0)+1.5y=180$
  • $2(0)=0$, so we get $1.5y=180$
  • divide by $1.5$ to get $y=120$
  • the y-intercept is $(0,120)$

  x-intercept

  • $2x+1.5(0)=180$
  • $1.5(0)=0$, so we get $2x=180$
  • divide by $2$ to get $x=90$
  • the x-intercept is at $(90,0)$

  Finally, draw a line through the two points to create a line that include all the possible solutions of the equation $2x+1.5y=180$.

• Decide how many marbles and colored pencils Sue can buy.

  Hints

  $x$ represents the number of colored pencils, and $y$ represents the number of marbles.

  Write the equation in standard form:

  $\begin{array}{c|c|c|c|c} A&x&B&y&C\\ 1&x&0.5&y&10. \end{array}$.

  The equation for this situation written in standard form: $x+0.5y=10$.

  Use a graph to verify each combination.

  Use this graph to verify the combinations.

  Solution

  Let's start by using the standard form of an equation:

  $Ax+By=C$.

  • $x$ represents the unknown number of colored pencils.
  • $y$ represents the unknown number of marbles.
  • $A=1$ is the price for one colored pencil.
  • $B=0.5$ is the price for one marble.
  • $C=10$ is the total amount of Sue's pocket money.

  With this information we can set up an equation in standard form:

  $x+0.5y=10$.

  The corresponding graph can be seen on the right side.

  Let's check the combinations of pencils and marbles:

  10 pencils and 5 marbles.

  The point $(10,5)$ doesn't lie on the line. This isn't a solution of the equation.

  $~$

  5 pencils and 10 marbles

  The point $(5,10)$ lies on the line. Let's calculate:

  $5+0.5\times 10=5+5=10$ $\surd$.

  $~$

  6 pencils and 8 marbles

  The point $(6,8)$ lies on the line, so it's a correct answer:

  $6+0.5\times 8=6+4=10$ $\surd$.

  $~$

  8 pencils and 6 marbles

  The point $(8,6)$ doesn't lie on the line, so it can't be an answer to the equation.

  $~$

  12 pencils and 4 marbles

  The point $(12,4)$ doesn't lie on the line.

  $~$

  4 pencils and 12 marbles

  The point $(4,12)$ lies on the line:

  $4+0.5\times 12=4+6=10$ $\surd$.

• Explain how you can determine the number of large scoops of ice cream.

  Hints

  We know the price and the number of the small scoops, but we don't know the number of the large scoops.

  Write an equation in standard form: $Ax+By=C$.

  To check your solution, you can substitute $y=5$ and $x$ into the equation written in standard form.

  Solution

  We already know the price of a large scoop: $\$1.50$

  The corresponding number of scoops is unknown. So let's assign $x$ to this number. So we can write the term:

  $1.5 \times x$ which represents the total cost of large scoops.

  For the small scoops, the price as well as the number of scoops are known. So we get the equation:

  $0.9\times 5=4.5$

  So the total costs for 5 small scoops of ice cream is $\$4.50$.

  Adding the two terms we get the expression:

  $1.5x+4.5$

  This represents the total cost for the ice cream. Because Sue has $\$9$ to spendl, we can write and solve the following equation:

  $1.5x+4.5=9$

  Subtract $4.5$ on both sides of the equation and rewrite as:

  $1.5x=4.5$

  Dividing both sides by $1.5$ is equal to the solution for x:

  $x=3$

  Therefore, Sue can buy three large scoops of ice cream for herself to eat.

• For standard form, determine all parts of the equation.

  Hints

  Use this pattern

  $\begin{array}{c|c|c|c|c} \text{price for lemonade}&\text{number of glasses}&\text{price for cookie}&\text{number of cookies}&\text{total}\\ 2&x&?&?&? \end{array}$

  You already know which variable is assigned to the number of glasses and which to the number of cookies.

  Use the following pattern:

  $\begin{array}{c|c|c|c|c} \text{price for lemonade}&\text{number of glasses}&\text{price for cookie}&\text{number of cookies}&\text{total}\\ A&x&B&y&C \end{array}$

  Solution

  An equation in standard form looks like this:

  $Ax+By=C$

  • $x$ is the unknown number of sold glasses of lemonade
  • $y$ is the unknown number of sold cookies
  • $A=2$ is the price for one glass of lemonade
  • $B=1.5$ is the price for one cookie
  • $C=180$ is the total amount Kayla earned that day

  We can write the equation in standard form:

  $2x+1.5y=180$

  Since we know that Kayla sold 80 cookies, we can substitute $y$ with $80$ then we can rewrite the equation:

  $2x+1.5\times 80=180$

• Decide which equation in standard form corresponds to the graph.

  Hints

  Using the standard form, there are many combinations of x and y to write the equation of a line.

  For this line the intercepts are at $(0,16)$ and $(20,0)$.

  Solution

  Write an equation in standard form, $Ax+By=C$, for the given line.

  Let's look at the intercepts, $(20,0)$ and $(0,16)$:

  • if we plug in $x=20$ and $y=0$, we get $20\times A=C$
  • for $(0,16)$ we get $16\times B=C$
  • now we choose any value for $C$, for example $C=320$
  • then we get $A=16$ and $B=20$
  • One possible equation is $16x+20y=320$

  Why can we choose any value for $C$? For any other value, we get an equation which is a multiple of the equation shown above. You don't believe that? Let's take a look at another example for $C=160$. Now we get $A=8$ and $B=10$, and the equation written in standard form is $8x+10y=160$.

  We can say that any multiple of the equation $16x+20y=320$ is the same line that is graphed on the right:

  • $0.8x+y=16$
  • $2x+2.5x=40$
  • and so on