Question

below are two separate examples of students work using a system of linear equations use the students work to answer the following questions

Answer 1

This question involves high school mathematics, specifically the formation and manipulation of linear equations to represent real-world situations. Students must identify independent and dependent variables and express their relationships through equations.

Step-by-step explanation:

The subject in question pertains to the use of linear equations in various contexts, reflecting the application of mathematics to real-life scenarios. In these problems, one typically defines an independent variable and a dependent variable, and establishes a relationship between the two through a linear equation of the form y = mx + b, where m represents the slope and b the y-intercept. The examples provided cover a range of situations, from calculating the total number of hours to complete a task based on square footage to understanding the relationship between the age of drivers and motor vehicle fatalities.

By solving these equations or creating them based on given information, students enhance their problem-solving skills and their understanding of how mathematical principles can be applied to analyze and interpret data. Additionally, these examples also highlight the importance of determining the correct independent and dependent variables in a study, for instance, using the age of drivers to predict the number of fatalities or the number of family members to understand changes in the weekly grocery bill.

Answer 2

Part A.

By setting x and y as the volumes of the two solutions that mixed form a 12% sugar solution of 50mL we can formulate a system of two linear equations that let us calculate x and y. The first equation states that the volumes of the solutions must add up to 50, then x + y = 50, as you can see, the student properly formulated this equation, the second one represents the amount of sugar in the 50mL solution. By multiplying the volume of each solution by the percentage of sugar, we get the amount of sugar in the solution, then, when we multiply 0.07 (7%) by x, we get the sugar in the first solution and 0.15 (15%) by y, we get the sugar in the second solution, this sugar must add up to 50×0.12 (the sugar in the final solution), then the equation that must be formulated is:

0.07x + 0.15y = 50×0.12

0.07x + 0.15y = 6

As you can see, there are two mistakes in the equation formulated by the student, the 0.7 in the first term and the number on the right side of the equation.

Then, the corrected system of equations is

x + y = 50

0.07x + 0.15y = 6

Part B.

There is one error in the student work, you can notice it in the first step "Multiplying the first equation by (-10)", the thing is that when you multiply an equation by a given number, you have to multiply both sides, even the terms appearing on the right of the "=", but the student didn't do that, he/she multiplied only the terms on the left side, and left the number on the right unchanged.

The student should get:

-10x - 10y = -1900