Problem 78 - Introductory Algebra for College Students

Question

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{l}2 x-3 y=7 \\ 3 x+5 y=1\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is the point of intersection of these two lines as graphed on a graphing utility. The exact point would depend on the particular graphing tool used, but it can be estimated visually.

Step 1: Converting Equations into Standard Form

The first step is to convert equations into y = f(x) form. \[ y = \frac{2x - 7}{3} \] for the first equation and \[ y = \frac{1 - 3x}{5} \] for the second one

Step 2: Graph the Equations

Now, graph these two equations using a graphing utility. For the first equation, it will be a straight line with slope 2/3 and intercept -7/3. For the second equation, it will also be a straight line with slope -3/5 and intercept 1/5.

Step 3: Determine the Point of Intersection

Identify the point of intersection of two lines, this intersection point is the solution to the system of equations. This point provides the values of 'x' and 'y' which satisfy both the equations simultaneously.

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is an integral part of algebra that helps in visualizing the solutions to a system of equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.

Linear equations can be graphed on a coordinate plane by finding at least two solutions for the equation and then drawing a line through these points. Typically, we start by finding the y-intercept (the point where the line crosses the y-axis) by setting the value of x to zero in the equation and solving for y. Next, we use the slope, which is the ratio of the change in y to the change in x between two points on the line, to determine another point.

A slope can be positive or negative, and it indicates how steep the line is and in which direction it runs. If students find it challenging to draw the graph by hand, using a graphing utility can simplify the task by accurately plotting the line and ensuring precision. It is crucial to get comfortable with graphing as it is not only foundational for algebra but also for higher-level math courses and various real-world applications.

Standard Form of Linear Equations

The standard form of a linear equation is a way of writing the equation that provides a clear view of the x- and y-intercepts. This form is expressed as \(Ax + By = C\), where A, B, and C are integers, and A is a nonnegative integer. When we work with linear equations in standard form, it makes it straightforward to compute the intercepts which are essential for graphing.

To graph an equation that is already in standard form, we find the x- and y-intercepts by letting y and then x equal zero and solving for the other variable. These intercepts give us two points through which we can draw our line. Moreover, converting an equation to standard form can often make it easier to identify parallel and perpendicular lines based on their coefficients.

Many students find standard form helpful because it avoids the need for fractions, which can sometimes complicate calculations. Especially when working with equations that have larger or compound numbers, standard form can simplify the problem and the graphing process.

Point of Intersection

The point of intersection is where two or more graphs meet or cross each other on the coordinate plane. In the context of a system of linear equations, it's the physical representation of the solution to the system. This specific point brings to life the pair of x and y values that satisfy both equations simultaneously.

To find the point of intersection using graphing, we can graph each equation on the same set of axes. Where the lines intersect is our point of interest. With technology such as graphing calculators or computer software, this process is more accurate and less time-consuming. Alternatively, one can solve the system algebraically using methods such as substitution or elimination to find the same point.

When two lines intersect, it means that the system of equations has one solution, which is called a consistent and independent system. If the lines are parallel and never intersect, the system is inconsistent and has no solution. When two lines are coincident, meaning they lie on top of one another, the system has infinitely many solutions and is described as consistent and dependent. Thus, understanding the point of intersection conceptually aids in analyzing the relationship between linear equations and can provide insights into the nature of their solutions.