Solve. See a Concept Check in this section. The number of...

Solve. See a Concept Check in this section. The number of men and women receiving bachelor's degrees each year has been steadily increasing. For the years 1970 through the projection of 2014 , the number of men receiving degrees (in thousands) is given by the equation $y=3.9 x+443$ and for women, the equation is $y=14.2 x+314$ where $x$ is the number of years after $1970 .$ (Source: National Center for Education Statistics) (IMAGE CANNOT COPY) A. Use the substitution method to solve this system of equations. (Round your final results to the nearest whole numbers.) B. Explain the meaning of your answer to part (a). C. Sketch a graph of the system of equations. Write a sentence describing the trends for men and women receiving bachelor degrees.

Solve. See a Concept Check in this section.
The number of men and women receiving bachelor's degrees each year has been steadily increasing. For the years 1970 through the projection of 2014 , the number of men receiving degrees (in thousands) is given by the equation $y=3.9 x+443$ and for women, the equation is $y=14.2 x+314$ where $x$ is the number of years after $1970 .$ (Source: National Center for Education Statistics) (IMAGE CANNOT COPY)
A. Use the substitution method to solve this system of equations. (Round your final results to the nearest whole numbers.)
B. Explain the meaning of your answer to part (a).
C. Sketch a graph of the system of equations. Write a sentence describing the trends for men and women receiving bachelor degrees.

Key Concepts

Interpretation of Intersection

The interpretation of the intersection point in a system of equations involves understanding what the common solution represents within a real-world context. It signifies the value at which two different trends or conditions become equal, thereby offering insights into the dynamics of the situation being modeled.

Graphical Representation

Graphical representation is the process of plotting equations on a coordinate plane to visually interpret their relationships. For systems of linear equations, the graph helps identify the intersection point, which represents the solution to the system, and provides a visual understanding of how the variables interact.

System of Linear Equations

System of linear equations consists of two or more equations that share the same set of variables. Solving such systems means finding the values of these common variables that simultaneously satisfy every equation. This concept is essential in mathematics for modeling situations where different conditions must hold at the same time.

Substitution Method

The substitution method is a technique used to solve systems of equations, where one equation is rearranged to express one variable in terms of another, and then this expression is substituted into the other equation. This reduces the system to a single equation with one unknown, simplifying the process of finding the solution.

Linear Modeling

Linear modeling involves representing real-world relationships with linear equations, where variables change at constant rates. This concept is fundamental in statistics and algebra, as it simplifies the analysis and prediction of trends over time or between groups.