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Write an Equation Given the Slope and a Point

Solving y=mx+b for 'b'

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Real World Applications – Algebra I


How can we represent how much a person earns making clothes in Bangladesh?

Student Exploration

Bangladesh is one of the most popular countries for outsourcing labor because the labor is so cheap. Let’s apply our knowledge of linear equations to represent this relationship.

A child makes $20 per month for in a garment factor. Since this is a rate, this is the slope for our equation. Let’s say he/she would work every day out of the 30 days of the month. Our slope would then be represented as \begin{align*}\frac{\$20}{30} \ days\end{align*}$2030 days, or \begin{align*}\frac{2}{3}\end{align*}23. Let’s also say that this child has started saving money for his family and after working for 3 weeks, he has $24 for his family.

From this example, we have both the slope \begin{align*}\left(\frac{2}{3}\right)\end{align*} and a point (21, 24). (The \begin{align*}x-\end{align*}value in the point (21, 24) comes from the fact that there are 21 days in 3 weeks. For this relationship, we must use days, since the slope includes days, not weeks.) We can create an equation with this information. Let’s use \begin{align*}y = mx + b\end{align*} and follow the steps given in the concept.

\begin{align*} y &= mx + b\\ 24 &= \left(\frac{2}{3}\right)(21) + b\\ 24 &= \frac{42}{3} + b\\ 24 &= 14 + b\\ 10 &= b\\ y &= \left(\frac{2}{3}\right)x + 10\end{align*}

In this equation, the “10,” or the “\begin{align*}b\end{align*}” value represents how much the child had before he started working. This equation is in slope-intercept form.

If we were to represent this equation in standard form, we first want to get the \begin{align*}x-\end{align*}term on the same side as the \begin{align*}y-\end{align*}term.

\begin{align*}\left(\frac{2}{3}\right)x + y = 10\end{align*}

We also want all whole numbers in our equation. Let’s multiply every term by “3” so the denominator disappears.

We now have \begin{align*}2x + 3y = 30\end{align*}

Extension Investigation

Try finding the equation of the line in both slope-intercept form and standard form with the following information.

The child’s family found out that their child started saving money and thought it was a good idea and started saving too. One parent decided that all of her earnings are going toward their savings, and the other parent’s earnings will go toward the family’s expenses. The mother earns $36 per month. After 60 days, the family has $82.

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