Solving Systems Of Equations Find The Y-Coordinate

Jul 12, 2025 by ADMIN 51 views

Solving Systems of Equations: Finding the Y-Coordinate

In the realm of mathematics, solving systems of equations is a fundamental skill. These systems, comprised of two or more equations with shared variables, often represent real-world scenarios where multiple conditions must be satisfied simultaneously. This article delves into the process of solving a specific system of linear equations, focusing on determining the y-coordinate of the solution. We'll explore the methods involved and provide a clear, step-by-step guide to tackle such problems.

The system of equations we'll be addressing is:

5x + 2y = 7
-2x + 6y = 9

Our goal is to find the values of x and y that satisfy both equations simultaneously. However, the problem specifically asks for the y-coordinate of the solution, rounded to the nearest tenth. This means we can strategically employ methods that prioritize finding y without necessarily solving for x first. Let's dive into the techniques we can use to solve this problem.

There are several established methods for solving systems of equations, each with its strengths and suitability for different situations. The two primary methods we'll consider here are:

Substitution: This method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with a single variable, which can then be easily solved.
Elimination (also known as Addition/Subtraction): This method aims to eliminate one variable by manipulating the equations so that the coefficients of that variable are opposites. Adding the equations then cancels out that variable, leaving a single equation with one variable.

For this particular problem, the elimination method might be slightly more efficient for directly finding the y-coordinate, as we can manipulate the equations to eliminate x. However, let's explore both methods to demonstrate their versatility and understand which one suits our needs best.

The elimination method shines when we want to target a specific variable. In this case, since we need the y-coordinate, we can strategically eliminate x. Here's how it works:

Identify the Target Variable: We want to eliminate x, so we focus on the coefficients of x in both equations: 5 and -2.
Find a Common Multiple: The least common multiple of 5 and 2 is 10. We'll aim to make the coefficients of x in both equations equal to 10 and -10.
Multiply the Equations:
- Multiply the first equation (5x + 2y = 7) by 2: This gives us 10x + 4y = 14.
- Multiply the second equation (-2x + 6y = 9) by 5: This gives us -10x + 30y = 45.

Add the Equations: Now we have:

10x + 4y = 14
-10x + 30y = 45

Adding these equations eliminates x:

(10x - 10x) + (4y + 30y) = 14 + 45
0x + 34y = 59
34y = 59

Solve for y: Divide both sides by 34:
```
y = 59 / 34
y ≈ 1.735
```
Round to the Nearest Tenth: Rounding 1.735 to the nearest tenth gives us 1.7.

Therefore, the y-coordinate of the solution, using the elimination method, is approximately 1.7.

Now, let's tackle the same system using the substitution method. While we are primarily interested in the y-coordinate, this method will illustrate an alternative approach.

Solve one equation for one variable: We can choose either equation and solve for either x or y. Let's solve the first equation (5x + 2y = 7) for x:
```
5x = 7 - 2y
x = (7 - 2y) / 5
```
Substitute: Substitute this expression for x into the second equation (-2x + 6y = 9):
```
-2 * ((7 - 2y) / 5) + 6y = 9
```
Simplify and Solve for y:
- Multiply both sides of the equation by 5 to eliminate the fraction:
```
-2(7 - 2y) + 30y = 45
```
- Distribute the -2:
```
-14 + 4y + 30y = 45
```
- Combine like terms:
```
34y = 59
```
- Divide both sides by 34:
```
y = 59 / 34
y ≈ 1.735
```
Round to the Nearest Tenth: Again, rounding 1.735 to the nearest tenth gives us 1.7.

The substitution method also yields a y-coordinate of approximately 1.7.

It's always a good practice to verify our solution by plugging the y-value back into either of the original equations and solving for x. Then, we can check if both x and y values satisfy both equations.

Using the value y ≈ 1.7, let's plug it into the first equation (5x + 2y = 7):

5x + 2(1.7) = 7
5x + 3.4 = 7
5x = 3.6
x = 3.6 / 5
x ≈ 0.72

Now, let's check if these values of x and y satisfy the second equation (-2x + 6y = 9):

-2(0.72) + 6(1.7) = 9
-1.44 + 10.2 = 9
8.76 ≈ 9

The result is close to 9, which confirms that our solution is accurate, considering we rounded the y-value. Small discrepancies can occur due to rounding, but the verification step assures us that our solution is correct to the nearest tenth.

In summary, we successfully solved the system of equations

5x + 2y = 7
-2x + 6y = 9

using both the elimination and substitution methods. Both approaches led us to the same y-coordinate of approximately 1.7, rounded to the nearest tenth. This demonstrates the power and flexibility of these methods in solving systems of linear equations.

The elimination method, in this case, proved to be slightly more direct in finding the y-coordinate, as we strategically eliminated x. However, the substitution method is equally valuable and applicable to a wide range of systems. The choice of method often depends on the specific structure of the equations and personal preference.

Understanding these methods is crucial for tackling various mathematical problems and real-world applications involving simultaneous conditions. By mastering these techniques, you'll be well-equipped to solve systems of equations and extract valuable information from them.

Key Takeaways:

Systems of equations can be solved using elimination or substitution.
The elimination method involves manipulating equations to cancel out a variable.
The substitution method involves solving one equation for one variable and substituting it into the other equation.
Always verify your solution by plugging the values back into the original equations.
Rounding can introduce small discrepancies, but the solution should be accurate to the specified decimal place.

By consistently practicing these methods and applying them to diverse problems, you can build a strong foundation in solving systems of equations, a valuable skill in mathematics and beyond.

To further solidify your understanding, try solving these systems of equations and find the y-coordinate of the solution (rounded to the nearest tenth):

3x - y = 5 x + 2y = 4
2x + 3y = 10 x - y = 1
4x + 5y = 12 -2x + y = 3

Good luck, and remember to practice regularly to master these techniques!