Equivalent Systems Of Equations A Comprehensive Guide

Understanding Systems of Equations is crucial for solving various mathematical problems. A system of equations involves two or more equations with the same variables. The solution to a system of equations is the set of values for the variables that satisfy all the equations simultaneously. In this article, we will explore how to determine which system of equations is equivalent to a given system. We will focus on the given system: $\left\{\begin{array}{r}y=9 x^2 \\ x+y=5\end{array}\right.$ and analyze the provided options to identify the equivalent system.

Analyzing the Given System

To begin, let's delve into the intricacies of the given system of equations. We have two equations:

1. y = 9x²: This equation represents a parabola. The variable y is expressed as a quadratic function of x. This means that the graph of this equation will be a U-shaped curve.
2. x + y = 5: This equation represents a linear relationship between x and y. It can be rewritten as y = 5 - x, which is the equation of a straight line with a slope of -1 and a y-intercept of 5.

To find the solution to this system, we need to find the points where the parabola and the line intersect. These intersection points represent the values of x and y that satisfy both equations. There are several methods to solve such systems, including substitution, elimination, and graphical methods. In this case, substitution appears to be the most straightforward approach. We can substitute the expression for y from the first equation into the second equation, or vice versa, to obtain a single equation in one variable. This single equation can then be solved using algebraic techniques.

Option 1: Substitution Method

Consider the first option: $\left\{\begin{aligned} 5-x & =9 x^2 \\ y & =5-x\end{aligned}\right.$

This system appears to be derived from the original system by substituting the expression for y from the second equation (x + y = 5) into the first equation (y = 9x²). Let's examine this process step-by-step:

1. From the second equation, we have x + y = 5. Solving for y, we get y = 5 - x.
2. Substituting this expression for y into the first equation (y = 9x²), we get 5 - x = 9x².
3. The second equation in the proposed system, y = 5 - x, is simply a rearrangement of the original second equation.

Therefore, this system is indeed equivalent to the original system. The first equation 5 - x = 9x² is a quadratic equation in x, which can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Once the values of x are found, they can be substituted back into the equation y = 5 - x to find the corresponding values of y. This approach provides a clear and logical way to solve the system by reducing it to a single variable equation.

Option 2: Analyzing the Alternative System

Now, let's examine the second option: $\left\{\begin{array}{l}y=9 y^2-90 y+225 \\\end{array}\right.$

This system consists of only one equation, which involves y as both a linear and quadratic term. This raises a significant concern about whether this system is equivalent to the original system, which has two equations involving both x and y. The given equation y = 9y² - 90y + 225 appears to be derived from a more complex substitution process, and it's not immediately clear how it relates to the original system.

To determine if this equation is equivalent, we would need to meticulously retrace the steps that might have led to this form. It's possible that a substitution was made from the equation x + y = 5 into y = 9x², but the resulting expression seems overly complicated. Let's attempt to derive this equation from the original system to check its validity:

1. From x + y = 5, we have x = 5 - y.
2. Substitute this into y = 9x² to get y = 9(5 - y)².
3. Expanding the square, we get y = 9(25 - 10y + y²).
4. This simplifies to y = 225 - 90y + 9y², which is the equation given in the second option.

However, having only one equation in y means that we can solve for y, but we will need to use one of the original equations to find the corresponding x values. This single equation, while mathematically derived from the original system, might not provide a complete and straightforward solution as the system in Option 1. The lack of a separate equation relating x and y directly makes it less convenient to find the solution set for the original system.

Detailed Comparison and Equivalence Assessment

To make a definitive determination of equivalence, we must evaluate whether each system provides the same solution set as the original system. We've already established that Option 1 is derived from a direct substitution, making it a strong candidate for equivalence. Option 2, while derived mathematically, presents a slightly more complex scenario due to the absence of an explicit equation for x in terms of y or vice versa.

Option 1: Detailed Analysis

The system in Option 1, $\left\{\begin{aligned} 5-x & =9 x^2 \\ y & =5-x\end{aligned}\right.$, is constructed by substituting y = 5 - x into the first original equation. This yields a quadratic equation in x, specifically 9x² + x - 5 = 0. Solving this quadratic equation will give us the possible x values. For each x value, we can then use the equation y = 5 - x to find the corresponding y values. This method directly addresses the solution set for both x and y, ensuring a clear path to the solution.

The quadratic equation 9x² + x - 5 = 0 can be solved using the quadratic formula: $x = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$ In this case, a = 9, b = 1, and c = -5. Plugging these values into the formula, we get $x = \frac{-1 \pm \sqrt{1^2 - 4(9)(-5)}2(9)} = \frac{-1 \pm \sqrt{1 + 180}}{18} = \frac{-1 \pm \sqrt{181}}{18}$ This yields two values for x $x_1 = \frac{-1 + \sqrt{181}{18} \approx 0.685$ $x_2 = \frac{-1 - \sqrt{181}}{18} \approx -0.796$ For each of these x values, we can find the corresponding y values using y = 5 - x:

• For x₁ ≈ 0.685, y₁ = 5 - 0.685 ≈ 4.315
• For x₂ ≈ -0.796, y₂ = 5 - (-0.796) ≈ 5.796

Thus, Option 1 allows us to find the solution set systematically.

Option 2: Detailed Analysis

Option 2 presents the equation y = 9y² - 90y + 225. This is a quadratic equation in y, which can be rewritten as 9y² - 91y + 225 = 0. While we can solve for y using the quadratic formula, we would still need to substitute each y value back into one of the original equations (e.g., x + y = 5) to find the corresponding x values. This adds an extra step compared to Option 1, where the y values are directly obtained from the x values.

Solving the quadratic equation 9y² - 91y + 225 = 0 using the quadratic formula: $y = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, a = 9, b = -91, and c = 225. Plugging these values in, we get $y = \frac{91 \pm \sqrt{(-91)^2 - 4(9)(225)}2(9)} = \frac{91 \pm \sqrt{8281 - 8100}}{18} = \frac{91 \pm \sqrt{181}}{18}$ The two values for y are $y_1 = \frac{91 + \sqrt{181}{18} \approx 5.796$ $y_2 = \frac{91 - \sqrt{181}}{18} \approx 4.315$ For each y value, we use x = 5 - y to find the corresponding x values:

• For y₁ ≈ 5.796, x₁ = 5 - 5.796 ≈ -0.796
• For y₂ ≈ 4.315, x₂ = 5 - 4.315 ≈ 0.685

Notice that these are the same solution pairs as we found with Option 1, but the process is slightly less direct.

Conclusion

After a thorough analysis, it is evident that both options are mathematically derived from the original system. However, Option 1 offers a more straightforward path to solving for x and then finding y, making it a more directly equivalent system. Option 2, while valid, requires an additional step of back-substitution to find the x values once the y values are determined. Therefore, the system in Option 1 is the most efficient and directly equivalent system to the original system of equations.

Option 1: $\left\{\begin{aligned} 5-x & =9 x^2 \\ y & =5-x\end{aligned}\right.$

This detailed analysis highlights the importance of understanding the relationships between equations in a system and how substitutions can be used to simplify and solve them. The most equivalent system is the one that most clearly and directly leads to the solution set.