Solving 2x + 3y = 2 And Y = 1/2x + 3 A Step-by-Step Guide

In mathematics, solving systems of equations is a fundamental concept with applications across various fields, from engineering and physics to economics and computer science. A system of equations is a set of two or more equations that involve the same variables. The solution to a system of equations is the set of values for the variables that satisfy all the equations simultaneously. This article will delve into solving a specific system of linear equations, providing a detailed step-by-step approach to finding the solution. We will explore the concepts behind the solution process, making it easier to understand and apply to other problems.

The system of equations we will be focusing on is:

$$\begin{array}{l} 2x + 3y = 2 \\ y = \frac{1}{2}x + 3 \end{array}$$

Our goal is to find the values of $x$ and $y$ that satisfy both equations. There are several methods to solve systems of equations, including substitution, elimination, and graphical methods. In this case, we will use the substitution method, as it is particularly well-suited for systems where one equation is already solved for one variable in terms of the other. Understanding these methods and their applications is crucial for mastering algebra and its practical applications.

Understanding the Equations

Before diving into the solution, let's take a moment to understand the equations we are working with. The first equation, $2x + 3y = 2$, is a linear equation in two variables. This means that its graph is a straight line in the coordinate plane. The coefficients 2 and 3 represent the slopes associated with the variables $x$ and $y$, respectively, and the constant term 2 represents the y-intercept after rearranging the equation into slope-intercept form. Linear equations are foundational in mathematics and represent relationships where the change in one variable is proportional to the change in another.

The second equation, $y = \frac{1}{2}x + 3$, is also a linear equation, but it is already in slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Here, the slope is $\frac{1}{2}$, and the y-intercept is 3. This form makes it easy to visualize the line's orientation and position on the coordinate plane. The slope-intercept form is a powerful tool for quickly understanding and graphing linear equations, making it easier to analyze their behavior and relationship to other equations.

The fact that we have two linear equations suggests that we are looking for the point where these two lines intersect on the coordinate plane. This point of intersection represents the solution to the system of equations, as it is the only point that lies on both lines and thus satisfies both equations simultaneously. Grasping the graphical representation of linear equations and their intersections can provide a deeper understanding of the algebraic solutions we calculate.

Method 1: Substitution Method

The substitution method is a powerful technique for solving systems of equations, especially when one equation is already solved for one variable in terms of the other. This method involves substituting the expression for one variable from one equation into the other equation, thereby eliminating one variable and resulting in a single equation with a single variable. This simplified equation can then be solved directly, and the solution can be substituted back into one of the original equations to find the value of the other variable. The substitution method is particularly effective when dealing with systems of linear equations, but it can also be applied to non-linear systems in certain cases.

In our system of equations,

$$\begin{array}{l} 2x + 3y = 2 \\ y = \frac{1}{2}x + 3 \end{array}$$

the second equation is already solved for $y$. This makes the substitution method a natural choice. We can substitute the expression for $y$ from the second equation into the first equation. This means replacing the $y$ in the first equation with the expression $\frac{1}{2}x + 3$. This substitution step is the core of the method, as it transforms the problem into a simpler form that can be easily solved.

Substituting $y = \frac{1}{2}x + 3$ into the first equation $2x + 3y = 2$, we get:

$$2x + 3(\frac{1}{2}x + 3) = 2$$

This equation now contains only the variable $x$, which we can solve for using algebraic techniques. The substitution method allows us to reduce a two-variable problem into a one-variable problem, making it a versatile tool in solving systems of equations. Once we find the value of $x$, we can then substitute it back into either of the original equations to find the corresponding value of $y$, completing the solution process.

Solving for x

Having substituted the expression for $y$ into the first equation, we now have a single equation in terms of $x$. Let's solve the equation:

$$2x + 3(\frac{1}{2}x + 3) = 2$$

First, we distribute the 3 across the terms inside the parentheses:

$$2x + \frac{3}{2}x + 9 = 2$$

Next, we combine the terms involving $x$. To do this, we need a common denominator for the coefficients of $x$, which is 2. So, we rewrite $2x$ as $\frac{4}{2}x$:

$$\frac{4}{2}x + \frac{3}{2}x + 9 = 2$$

Now, we can add the $x$ terms:

$$\frac{7}{2}x + 9 = 2$$

To isolate the $x$ term, we subtract 9 from both sides of the equation:

$$\frac{7}{2}x = 2 - 9$$

$$\frac{7}{2}x = -7$$

Finally, to solve for $x$, we multiply both sides by the reciprocal of $\frac{7}{2}$, which is $\frac{2}{7}$:

$$x = -7 \cdot \frac{2}{7}$$

$$x = -2$$

So, we have found that $x = -2$. This is a crucial step in solving the system of equations. With the value of $x$ determined, we can now substitute it back into one of the original equations to find the value of $y$. Solving for $x$ is often the most challenging part of solving systems of equations, and it requires careful algebraic manipulation to avoid errors. Once $x$ is found, the rest of the solution process typically involves straightforward substitution and calculation.

Solving for y

Now that we have found the value of $x = -2$, we can substitute it into one of the original equations to solve for $y$. The second equation, $y = \frac{1}{2}x + 3$, is more straightforward for this purpose because $y$ is already isolated. Substituting $x = -2$ into this equation, we get:

$$y = \frac{1}{2}(-2) + 3$$

First, we multiply $\frac{1}{2}$ by -2:

$$y = -1 + 3$$

Then, we add -1 and 3:

$$y = 2$$

So, we have found that $y = 2$. This completes the solution process. We now have both the $x$ and $y$ values that satisfy the system of equations. Substituting the value of $x$ back into the equation to find $y$ is a common technique in solving systems of equations, and it is particularly easy when one of the equations is already solved for one variable in terms of the other. This step ensures that we find the corresponding $y$ value for the $x$ value we calculated, leading us to the complete solution.

The Solution

We have found that $x = -2$ and $y = 2$. Therefore, the solution to the system of equations is the ordered pair $(-2, 2)$. This ordered pair represents the point where the two lines described by the equations intersect on the coordinate plane. The solution is the unique point that satisfies both equations simultaneously. Understanding that the solution is a point of intersection is crucial for visualizing and interpreting the solutions of systems of equations.

To verify our solution, we can substitute the values of $x$ and $y$ back into both original equations to ensure that they hold true. This is a good practice to catch any potential errors in our calculations. Let's check the first equation, $2x + 3y = 2$:

$$2(-2) + 3(2) = -4 + 6 = 2$$

The first equation is satisfied. Now, let's check the second equation, $y = \frac{1}{2}x + 3$:

$$2 = \frac{1}{2}(-2) + 3 = -1 + 3 = 2$$

The second equation is also satisfied. Since the values $x = -2$ and $y = 2$ satisfy both equations, we can confidently conclude that $(-2, 2)$ is the correct solution to the system of equations. Verifying the solution is an important step in the problem-solving process, as it ensures accuracy and reinforces our understanding of the solution concept.

Answer

The solution to the system of equations is:

A. (-2, 2)

Conclusion

In this article, we have demonstrated a step-by-step method to solve a system of linear equations using the substitution method. Solving systems of equations is a crucial skill in mathematics, with applications in various fields. The key steps include substituting one equation into another, solving for one variable, and then substituting the value back to find the other variable. We also emphasized the importance of verifying the solution to ensure accuracy. By understanding these concepts and practicing the techniques, you can confidently solve a wide range of systems of equations.

The substitution method, as demonstrated here, is just one of several methods for solving systems of equations. Other methods, such as the elimination method and graphical methods, can also be used, and the choice of method often depends on the specific equations in the system. Each method has its strengths and weaknesses, and familiarity with all of them can make you a more versatile problem solver. Furthermore, the ability to solve systems of equations is a gateway to more advanced mathematical topics, such as linear algebra and calculus, where systems of equations appear frequently.