Respuesta :
Solution:
A linear relationship is expressed as
[tex]y=ax+b\text{ ----- equation \lparen *\rparen}[/tex]let
[tex]\begin{gathered} y\Rightarrow cost\text{ of production} \\ x\Rightarrow number\text{ of cups of coffee} \end{gathered}[/tex]Given that the cost to produce 150 cups of coffee is $80, this implies that
[tex]\begin{gathered} 80=a(150)+b \\ \Rightarrow150a+b=80\text{ ---- equation 1} \end{gathered}[/tex]if the cost to produce 300 cups is $155, this implies that
[tex]\begin{gathered} 155=a(300)+b \\ \Rightarrow300a+b=155\text{ ---- equation 2} \end{gathered}[/tex]A) write a linear equation that expresses the cost why in terms of the number of cups of coffee x
To express the linear equation, we solve for a and b simultaneously.
By the method of elimination,
step 1: Subtract equation 1 from equation 2.
Thus,
[tex]\begin{gathered} (300a-150a)+(b-b)=(155-80) \\ \Rightarrow150a=75 \\ divide\text{ both sides by the coefficient of a.} \\ the\text{ coefficient of a is 150.} \\ thus, \\ \frac{150a}{150}=\frac{75}{150} \\ \Rightarrow a=0.5 \end{gathered}[/tex]step 2: Substitute the obtained value of a into equation 1.
Thus, from equation 1,
[tex]\begin{gathered} 150a+b=155 \\ where \\ a=0.5, \\ we\text{ have} \\ 150(0.5)+b=155 \\ \Rightarrow75+b=155 \\ subtract\text{ 75 from both sides of the equation,} \\ 75-75+b=155-75 \\ \Rightarrow b=80 \end{gathered}[/tex]Step 3: Substitute the value of a and b into equation (*).
Thus, the linear equation is expressed as
[tex]y=0.5x+80\text{ ----- equation 3}[/tex]B) Number of cups produced if the cost of production is 210.
To evaluate the number of cups, substitute the value of 210 for y in equation 3.
Thus, from equation 3,
[tex]\begin{gathered} y=0.5x+80 \\ where\text{ y=210} \\ thus, \\ 210=0.5x+80 \\ subtract\text{ 80 from both sides of the equation} \\ 210-80=0.5x+80-80 \\ \Rightarrow130=0.5x \\ divide\text{ both sides by 0.5} \\ \frac{130}{0.5}=\frac{0.5x}{0.5} \\ \Rightarrow x=260 \end{gathered}[/tex]Hence, the number of cups of coffee produced is 260.
