Solving Equations: Find The Ordered Pair (x, Y)

In this article, we'll dive into how to find the ordered pair (x, y) that satisfies two linear equations. Specifically, we'll use a table of values to determine where the equations y = -3x + 4 and y = 5x + 1 intersect. This is a common problem in mathematics, and understanding how to solve it can be incredibly useful.

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have two equations:

1. y = -3x + 4
2. y = 5x + 1

We need to find a pair of values (x, y) that makes both equations true simultaneously. In other words, we're looking for the point where the two lines represented by these equations intersect on a graph. The table provides us with some pre-calculated values, which should help us pinpoint the solution more easily.

Analyzing the Table

The table given in the problem provides values of x and corresponding values of y for each equation:

We are looking for a row where the y values for both equations are the same. By inspecting the table, we can see that at x = 0.3, y = -3x + 4 is 3.1 and y = 5x + 1 is 2.5. At x = 0.4, y = -3x + 4 is 2.8 and y = 5x + 1 is 3. It seems the intersection point lies somewhere between x = 0.3 and x = 0.4. However, none of the y values match exactly in the table.

Let's further analyze the equations to find when the y-values are equal. To find the exact ordered pair, we need to solve the equations simultaneously. Since both equations are solved for y, we can set them equal to each other:

-3x + 4 = 5x + 1

Solving the Equations Algebraically

Now, let's solve for x:

1. Add 3x to both sides: 4 = 5x + 3x + 1 4 = 8x + 1
2. Subtract 1 from both sides: 3 = 8x
3. Divide both sides by 8: x = 3/8 x = 0.375

Now that we have the value of x, we can plug it back into either equation to find the value of y. Let's use the first equation:

y = -3x + 4 y = -3(0.375) + 4 y = -1.125 + 4 y = 2.875

So, the ordered pair that satisfies both equations is (0.375, 2.875).

Verifying the Solution

To be absolutely sure, let's plug these values into the second equation as well:

y = 5x + 1 y = 5(0.375) + 1 y = 1.875 + 1 y = 2.875

The value of y is the same for both equations, so our solution is correct. Therefore, the ordered pair (0.375, 2.875) satisfies both equations.

Why This Matters

Understanding how to solve systems of equations is a fundamental skill in mathematics. It has applications in various fields, including physics, engineering, economics, and computer science. Being able to find the intersection point of two lines, or more generally, the solution to a set of equations, allows you to model and solve real-world problems. For example, in economics, you might use simultaneous equations to find the equilibrium point where supply and demand curves intersect.

Tips for Solving Similar Problems

Here are some tips to keep in mind when solving similar problems:

1. Understand the Problem: Make sure you know what you're being asked to find. In this case, it was the ordered pair that satisfies both equations.
2. Use the Table Wisely: The table can give you a good starting point. Look for values that are close to each other and try to narrow down the range where the solution might lie.
3. Solve Algebraically: If the table doesn't give you an exact answer, use algebraic methods to solve the equations simultaneously.
4. Verify Your Solution: Always plug your solution back into the original equations to make sure it works.
5. Graphing: Visualizing the equations as lines on a graph can help you understand the problem better. The intersection point of the lines is the solution you're looking for.

Common Mistakes to Avoid

• Arithmetic Errors: Be careful when performing arithmetic operations, especially when dealing with decimals or fractions. Double-check your calculations to avoid mistakes.
• Incorrect Substitution: Make sure you substitute the values correctly when verifying your solution. It's easy to mix up the x and y values, so pay attention to what you're doing.
• Misinterpreting the Table: Don't assume that the solution is in the table if you don't see an exact match. The table is just a guide, and the solution might lie between the values given.

Real-World Applications

The ability to solve systems of equations is not just a theoretical exercise. It has numerous real-world applications.

Economics

In economics, supply and demand curves can be represented by linear equations. The equilibrium point, where the supply and demand curves intersect, determines the market price and quantity of a product. Solving the equations simultaneously gives you the equilibrium price and quantity.

Engineering

In engineering, systems of equations are used to analyze circuits, design structures, and model fluid flow. For example, Kirchhoff's laws in circuit analysis can be expressed as a set of linear equations. Solving these equations gives you the currents and voltages in the circuit.

Computer Graphics

In computer graphics, systems of equations are used to perform transformations such as scaling, rotation, and translation. These transformations are represented by matrices, and solving the equations allows you to manipulate objects in 3D space.

Conclusion

In summary, we found the ordered pair (x, y) that satisfies both equations y = -3x + 4 and y = 5x + 1 by analyzing a table of values and solving the equations algebraically. The solution is (0.375, 2.875). Understanding how to solve systems of equations is a valuable skill with applications in various fields. Remember to use the table as a guide, solve algebraically, and verify your solution. With practice, you'll become more confident in solving these types of problems.

For further learning on systems of equations, check out this resource: Khan Academy - Systems of Equations