Solving Simultaneous Equations And Simplifying Surds A Step By Step Guide

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In mathematics, the ability to solve simultaneous equations and simplify surds is a fundamental skill. These techniques are essential not only for academic success but also for various applications in science, engineering, and other fields. This comprehensive guide will walk you through the process of simplifying simultaneous equations and surds, providing clear explanations and step-by-step solutions.

Solving Simultaneous Equations

Simultaneous equations, also known as systems of equations, are a set of two or more equations containing the same variables. The goal is to find the values of the variables that satisfy all equations in the system. There are several methods for solving simultaneous equations, including substitution, elimination, and graphing. In this guide, we will focus on the elimination method, which is particularly useful for linear equations.

The elimination method involves manipulating the equations in the system so that the coefficients of one of the variables are opposites. This allows us to eliminate that variable by adding the equations together. The resulting equation will contain only one variable, which can be easily solved. Once we have the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable.

Example Problems

Let's dive into some examples to illustrate the process of solving simultaneous equations using the elimination method.

a) Solving $2x + 5y = 31$ and $2x + 3y = 20$

In this system, we have two equations:

  1. 2x+5y=312x + 5y = 31

  2. 2x+3y=202x + 3y = 20

Notice that the coefficients of x in both equations are the same (2). To eliminate x, we can subtract the second equation from the first equation:

(2x + 5y) - (2x + 3y) = 31 - 20$ $2x + 5y - 2x - 3y = 11$ $2y = 11

Now, we can solve for y:

y=112=5.5y = \frac{11}{2} = 5.5

Next, substitute the value of y back into either equation 1 or equation 2. Let's use equation 2:

2x + 3(5.5) = 20$ $2x + 16.5 = 20$ $2x = 20 - 16.5$ $2x = 3.5$ $x = \frac{3.5}{2} = 1.75

Therefore, the solution to this system of equations is x = 1.75 and y = 5.5.

b) Solving $2x + 4y = 42$ and $6x - 4y = 30$

Here, we have the following system of equations:

  1. 2x+4y=422x + 4y = 42

  2. 6x−4y=306x - 4y = 30

In this case, the coefficients of y are opposites (4 and -4). We can eliminate y by adding the two equations together:

(2x + 4y) + (6x - 4y) = 42 + 30$ $2x + 4y + 6x - 4y = 72$ $8x = 72

Now, solve for x:

x=728=9x = \frac{72}{8} = 9

Substitute the value of x back into either equation 1 or equation 2. Let's use equation 1:

2(9) + 4y = 42$ $18 + 4y = 42$ $4y = 42 - 18$ $4y = 24$ $y = \frac{24}{4} = 6

Thus, the solution for this system of equations is x = 9 and y = 6.

c) Solving $-4x + 3y = -7$ and $y = 5x - 17$

We are given the following system:

  1. −4x+3y=−7-4x + 3y = -7

  2. y=5x−17y = 5x - 17

This time, we can use the substitution method since the second equation is already solved for y. Substitute the expression for y from equation 2 into equation 1:

-4x + 3(5x - 17) = -7$ $-4x + 15x - 51 = -7$ $11x - 51 = -7$ $11x = -7 + 51$ $11x = 44$ $x = \frac{44}{11} = 4

Now, substitute the value of x back into equation 2 to find y:

y = 5(4) - 17$ $y = 20 - 17$ $y = 3

Therefore, the solution to this system of equations is x = 4 and y = 3.

Simplifying Surds

Surds are irrational numbers that can be expressed as the root of a whole number. Simplifying surds involves expressing them in their simplest form by factoring out any perfect square factors from the radicand (the number under the root). This process makes it easier to work with surds in calculations and comparisons.

Basic Principles

The key principle in simplifying surds is to identify perfect square factors within the radicand. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). We can use the following property to simplify surds:

ab=aâ‹…b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}

This property allows us to separate the radicand into factors and simplify the square root of any perfect square factors.

Example Problems

Let's illustrate the simplification of surds with some examples.

a) Simplify ...

(The original query is incomplete and does not contain the surd to be simplified. Please provide the surd expression to be simplified for a complete solution.)

For example, if the surd to simplify was $\sqrt{75}$, the solution would be as follows:

To simplify $\sqrt{75}$, we first find the prime factorization of 75:

75=3×25=3×5275 = 3 \times 25 = 3 \times 5^2

Now, rewrite the surd using the prime factorization:

75=3×52\sqrt{75} = \sqrt{3 \times 5^2}

Using the property $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get:

75=3â‹…52\sqrt{75} = \sqrt{3} \cdot \sqrt{5^2}

Since $\sqrt{5^2} = 5$, we have:

75=53\sqrt{75} = 5\sqrt{3}

Thus, the simplified form of $\sqrt{75}$ is $5\sqrt{3}$.

Steps to Simplify Surds:

  1. Find the prime factorization of the radicand: Break down the number under the square root into its prime factors.
  2. Identify perfect square factors: Look for factors that are perfect squares (e.g., 4, 9, 16, 25, etc.).
  3. Rewrite the surd: Express the surd as the product of the square roots of the perfect square factors and the remaining factors.
  4. Simplify the square roots of perfect squares: Take the square root of each perfect square factor and write it outside the square root symbol.
  5. Combine the terms: Multiply any whole numbers outside the square root symbol and leave the remaining factors under the square root symbol.

Additional Tips for Solving Simultaneous Equations

  • Check your solutions: After finding the values of the variables, substitute them back into the original equations to verify that they satisfy both equations.
  • Choose the most efficient method: The elimination method is often the most straightforward for linear equations, but the substitution method may be more convenient in certain cases.
  • Be careful with signs: Pay close attention to the signs of the coefficients when manipulating the equations.
  • Practice regularly: The more you practice solving simultaneous equations, the more comfortable and confident you will become.

Additional Tips for Simplifying Surds

  • Memorize common perfect squares: Knowing the perfect squares up to a certain point (e.g., 144 or 225) can help you quickly identify perfect square factors.
  • Use prime factorization: Prime factorization is a powerful tool for finding all the factors of a number, including perfect square factors.
  • Simplify step by step: Break down the simplification process into smaller steps to avoid errors.
  • Practice regularly: Simplification of surds becomes easier with practice.

By mastering these techniques, you will enhance your mathematical skills and gain a deeper understanding of how to work with simultaneous equations and surds. Remember, practice is key to success, so work through a variety of problems to solidify your understanding.

Conclusion

Solving simultaneous equations and simplifying surds are fundamental skills in mathematics. This guide has provided a comprehensive overview of the techniques involved, including the elimination and substitution methods for simultaneous equations and the process of simplifying surds by factoring out perfect squares. By understanding these concepts and practicing regularly, you can confidently tackle a wide range of mathematical problems. Remember to always check your solutions and consider the most efficient method for each problem. With dedication and practice, you can master these essential mathematical skills.