Problem 8 - College Algebra

Question

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &2 x-3 y=-13\\\ &y=2 x+7 \end{aligned} $$

Step-by-Step Solution

Answer

The solutions for x and y are \(-2\) and \(3\) respectively, hence, the solution to the system of equations is \((-2,3)\)

Step 1: Substitute y in the first equation

Given the two equations: \(2x-3y = -13\) and \(y = 2x+7\), substitute \(y\) in the first equation with \(2x+7\), resulting in: \(2x - 3(2x+7) = -13\)

Step 2: Simplify the equation

Distribute the \(3\) within the parentheses: \(2x - 6x - 21 = -13\). Combine like terms to simplify the equation: \(-4x - 21 = -13\)

Step 3: Isolate the variable x

Add \(21\) to both sides to isolate the variable on one side: \(-4x = -13 + 21\). This results in \(-4x = 8\). Divide by \(-4\) to solve for x, obtaining \(x = -2\)

Step 4: Substitute x in the second equation

Substitute \(x = -2\) in the second equation \(y = 2x + 7\), giving \(y = 2(-2) + 7\). This simplifies to \(y = -4 + 7\) thus, \(y = 3\)

Key Concepts

Systems of Equations

A system of equations is essentially a set of two or more equations that share two or more variables. These equations are interrelated and together, they depict a particular situation or condition. When you solve a system of equations, you are finding the values of the variables that satisfy all equations simultaneously.
There are several methods to solve systems of equations:

• Graphical method
• Substitution method
• Elimination method

In our example, we used the substitution method where one equation is solved for one variable, and that expression is substituted into the other equation. This is particularly effective when one of the equations is already solved or easily solved for one variable.
This method helps in clearly finding precise values for variables, which might not always be easy to perceive graphically. Knowing when to use the substitution method is key; it's often more advantageous when one equation is already solved for a variable, as in our example: \( y = 2x + 7 \).

Linear Equations

A linear equation is an equation that makes a straight line when graphed. Linear equations have no variables with exponents other than one, no products of variables, and no variables in the denominator. The general form of a linear equation in two variables is \( ax + by = c \).
In our exercise, both equations are linear:

• \( 2x - 3y = -13 \)
• \( y = 2x + 7 \)

These are both linear because they form straight lines on a graph. Linear equations are fundamental in algebra and are used to describe various real-life situations, such as calculating expenses, profits, or motion. Understanding the structure and solving of linear equations makes solving problems in engineering, physics, and economics possible.

Solving Algebraic Equations

The process of solving algebraic equations is about finding what values of the variables make the equation true. Typically, this involves "isolating" the variable on one side of the equation to find its value.
Here's a simple breakdown involved in the steps we followed in our original solution:

• First, substituting a known value/equation directly helps simplify down variables into a single equation: converted the equation \( 2x - 3y = -13 \) with \( y = 2x+7 \).
• Next, simplifying involves distributing and combining like terms to make an easy equation to handle.
• Finally, isolate the desired variable by performing basic arithmetic, as done when solving for \( x \) and \( y \) in the example.

Solutions can often be checked by putting the values back into the original equations to ensure they hold true. This gives a reality-check that confirms the equations were interpreted and solved correctly. Solving equations builds a core algebraic skill allowing for logical reasoning and problem-solving in complex scenarios.