Solving a System of Linear Equations: Understanding and Solving 3x + 4y = 10 and 2x + 3y = 7

Linear equations are fundamental in mathematics and have various real-world applications. In this discussion, we will explore the system of linear equations:

3x + 4y = 10

2x + 3y = 7

We will go through the process of understanding, solving, and interpreting the solutions to this system of equations.

Let’s begin by understanding the two linear equations. These equations describe the relationship between two variables, x and y.

The first equation, 3x + 4y = 10, can be interpreted as follows: “Three times the value of x plus four times the value of y equals 10.”

The second equation, 2x + 3y = 7, can be interpreted as: “Two times the value of x plus three times the value of y equals 7.”

These equations represent two lines in a two-dimensional coordinate system. The goal is to find values of x and y that satisfy both equations simultaneously, meaning they are the coordinates of a point where the two lines intersect.

## Part 2: Solving the System (Approx. 300 words)

There are multiple methods to solve a system of linear equations, including substitution, elimination, and graphing. Here, we will use the elimination method.

## Step 1: Multiply Equations

We can start by manipulating the equations to make the coefficients of x or y in one of the equations equal in magnitude but opposite in sign to those in the other equation. This will allow us to eliminate one of the variables when we add or subtract the equations.

Let’s multiply the first equation by 2 and the second equation by 3 to make the coefficients of x equal:

(2 * 3x) + (2 * 4y) = (2 * 10) → 6x + 8y = 20

(3 * 2x) + (3 * 3y) = (3 * 7) → 6x + 9y = 21

## Step 2: Subtract Equations

Now, subtract the second equation from the first to eliminate x:

(6x + 8y) – (6x + 9y) = 20 – 21

This simplifies to:

-x – y = -1

## Step 3: Solve for y

Now that we have a single equation in terms of y, we can isolate y:

-x – y = -1

Add x to both sides:

-y = x – 1

Multiply both sides by -1 to isolate y:

y = -x + 1

## Step 4: Substitute y into one of the original equations

Now that we have an expression for y in terms of x, we can substitute this expression into one of the original equations to solve for x. Let’s use the first equation:

3x + 4y = 10 3x + 4(-x + 1) = 10

Simplify:

3x – 4x + 4 = 10

Combine like terms:

-x + 4 = 10

Subtract 4 from both sides:

-x = 6

Multiply both sides by -1:

x = -6

## Step 5: Find y

Now that we have found x, we can find y using the expression we derived earlier:

y = -x + 1 y = -(-6) + 1 y = 6 + 1 y = 7

## Part 3: Interpreting the Solutions (Approx. 150 words)

We have found the solutions to the system of equations:

x = -6 y = 7

These values represent the coordinates of the point where the two lines defined by the original equations intersect. In other words, they are the values of x and y that make both equations true simultaneously.

Geometrically, this means that the lines 3x + 4y = 10 and 2x + 3y = 7 intersect at the point (-6, 7) on a Cartesian plane.

This point of intersection is the solution to the system of equations. It is the unique pair of values for x and y that satisfy both equations. The solution represents a specific point in space where the relationships described by both equations hold true.