Matching Equations To Solutions Solving Linear Equations

Linear equations are fundamental in mathematics, forming the basis for more advanced algebraic concepts. These equations involve variables raised to the first power and can be solved to find the value of the unknown variable. Solving linear equations involves isolating the variable on one side of the equation by performing the same operations on both sides. This ensures the equation remains balanced, maintaining the equality. The solutions to these equations are the values that make the equation true.

To master solving linear equations, it's essential to understand the properties of equality. These properties allow us to manipulate equations while preserving their balance. The addition property of equality states that adding the same value to both sides of an equation does not change the solution. Similarly, the subtraction property of equality allows us to subtract the same value from both sides. The multiplication and division properties of equality dictate that multiplying or dividing both sides by the same non-zero value also preserves the equality. By applying these properties systematically, we can isolate the variable and determine its value. For instance, in the equation n - 13 = -12, we use the addition property of equality by adding 13 to both sides to find the value of n.

Moreover, understanding inverse operations is crucial for solving linear equations efficiently. Inverse operations are operations that undo each other. Addition and subtraction are inverse operations, as are multiplication and division. When solving an equation, we use inverse operations to isolate the variable. For example, to undo subtraction, we add; to undo multiplication, we divide. This approach simplifies the process of finding the solution, enabling us to tackle more complex equations with confidence. Practicing with various types of linear equations, including those involving fractions and negative numbers, helps solidify these concepts and builds proficiency in algebra. Linear equations are not just abstract mathematical constructs; they have real-world applications in various fields, from science and engineering to economics and finance.

Matching Equations to Solutions: A Step-by-Step Guide

In this section, we will delve into the specific task of matching linear equations with their correct solutions. This exercise not only reinforces the concepts of solving equations but also enhances problem-solving skills. The key to accurately matching equations with their solutions lies in systematically solving each equation and comparing the result with the given options. Let's consider the first equation: n - 13 = -12. To solve for n, we need to isolate the variable by undoing the subtraction. We add 13 to both sides of the equation:

n - 13 + 13 = -12 + 13 n = 1

Therefore, the solution to the equation n - 13 = -12 is n = 1. This process illustrates the application of the addition property of equality. Next, let's examine the second equation: n + 15 = -10. Here, the variable n is being added to 15. To isolate n, we need to undo the addition by subtracting 15 from both sides of the equation:

n + 15 - 15 = -10 - 15 n = -25

Thus, the solution to the equation n + 15 = -10 is n = -25. This demonstrates the use of the subtraction property of equality. Moving on to the third equation, n/5 = -1/5, we encounter a variable being divided by 5. To solve for n, we must undo the division by multiplying both sides of the equation by 5:

(n/5) * 5 = (-1/5) * 5 n = -1

Hence, the solution to the equation n/5 = -1/5 is n = -1. This highlights the multiplication property of equality. Finally, let's tackle the fourth equation: -5n = 1. In this case, the variable n is being multiplied by -5. To isolate n, we need to undo the multiplication by dividing both sides of the equation by -5:

-5n / -5 = 1 / -5 n = -1/5

Therefore, the solution to the equation -5n = 1 is n = -1/5. This exemplifies the division property of equality. By solving each equation step-by-step and carefully applying the properties of equality, we can accurately match each equation with its corresponding solution. This methodical approach is crucial for success in algebra and beyond.

Solving $n - 13 = -12$

The first equation we need to solve is $n - 13 = -12$. In this equation, we are trying to find the value of n that, when 13 is subtracted from it, results in -12. To isolate n, we need to undo the subtraction of 13. The inverse operation of subtraction is addition, so we will add 13 to both sides of the equation. This ensures that we maintain the balance of the equation, as any operation performed on one side must also be performed on the other side to keep the equation true.

Adding 13 to both sides, we get:

$n - 13 + 13 = -12 + 13$

The -13 and +13 on the left side cancel each other out, leaving us with n by itself. On the right side, -12 + 13 equals 1. Therefore, the equation simplifies to:

n = 1

This means that the value of n that satisfies the equation $n - 13 = -12$ is 1. To verify this solution, we can substitute 1 back into the original equation:

$1 - 13 = -12$

This is indeed true, confirming that our solution is correct. This step-by-step process of adding 13 to both sides demonstrates a fundamental principle in solving linear equations: using inverse operations to isolate the variable. This method can be applied to various equations, making it a crucial skill in algebra. Understanding and mastering this technique allows for the efficient and accurate solution of more complex equations. Furthermore, the ability to check the solution by substituting it back into the original equation provides an additional layer of confidence in the correctness of the answer.

Solving $n + 15 = -10$

Next, let's consider the equation $n + 15 = -10$. Here, we need to find the value of n that, when added to 15, gives us -10. To isolate n, we must undo the addition of 15. The inverse operation of addition is subtraction, so we will subtract 15 from both sides of the equation. Subtracting the same value from both sides ensures that the equation remains balanced and the equality is maintained.

Subtracting 15 from both sides, we have:

$n + 15 - 15 = -10 - 15$

On the left side, the +15 and -15 cancel each other out, leaving n isolated. On the right side, -10 - 15 equals -25. Thus, the equation simplifies to:

n = -25

This indicates that the value of n that satisfies the equation $n + 15 = -10$ is -25. To verify our solution, we can substitute -25 back into the original equation:

$-25 + 15 = -10$

This statement is true, confirming that our solution is accurate. This method of subtracting 15 from both sides highlights another crucial aspect of solving linear equations: utilizing inverse operations to isolate the variable. This technique is broadly applicable and forms the foundation for solving a wide range of algebraic problems. The process of verification, where the solution is substituted back into the original equation, is an essential step in ensuring the correctness of the answer and building confidence in one's problem-solving abilities.

Solving $\frac{n}{5} = -\frac{1}{5}$

Now, let's tackle the equation $\frac{n}{5} = -\frac{1}{5}$. In this equation, n is divided by 5, and we need to find the value of n that makes the equation true. To isolate n, we must undo the division. The inverse operation of division is multiplication, so we will multiply both sides of the equation by 5. This maintains the balance of the equation, ensuring that the equality holds.

Multiplying both sides by 5, we get:

$\frac{n}{5} * 5 = -\frac{1}{5} * 5$

On the left side, multiplying $\frac{n}{5}$ by 5 cancels out the division by 5, leaving n isolated. On the right side, multiplying $-\frac{1}{5}$ by 5 results in -1. Therefore, the equation simplifies to:

n = -1

This indicates that the value of n that satisfies the equation $\frac{n}{5} = -\frac{1}{5}$ is -1. To verify this solution, we can substitute -1 back into the original equation:

$\frac{-1}{5} = -\frac{1}{5}$

This statement is true, confirming that our solution is correct. This example demonstrates the importance of using the multiplication property of equality to solve equations involving division. The ability to efficiently and accurately solve such equations is a key skill in algebra. Moreover, the verification step reinforces the understanding that the solution must satisfy the original equation, providing a check for potential errors and enhancing problem-solving confidence.

Solving $-5n = 1$

Finally, let's address the equation -5n = 1. In this equation, n is being multiplied by -5, and our goal is to find the value of n that makes the equation true. To isolate n, we need to undo the multiplication by -5. The inverse operation of multiplication is division, so we will divide both sides of the equation by -5. Dividing both sides by the same value ensures that the equation remains balanced and the equality is maintained.

Dividing both sides by -5, we have:

$\frac{-5n}{-5} = \frac{1}{-5}$

On the left side, dividing -5n by -5 cancels out the multiplication by -5, leaving n isolated. On the right side, dividing 1 by -5 results in $-\frac{1}{5}$. Thus, the equation simplifies to:

n = -\frac{1}{5}

This means that the value of n that satisfies the equation -5n = 1 is $-\frac{1}{5}$. To verify our solution, we can substitute $-\frac{1}{5}$ back into the original equation:

$-5 * (-\frac{1}{5}) = 1$

This statement is true, confirming that our solution is accurate. This example underscores the significance of using the division property of equality to solve equations involving multiplication. It also highlights the importance of being careful with signs, especially when dealing with negative numbers. The ability to solve equations of this type is crucial for success in algebra, and the verification step ensures that the solution is correct and reinforces the understanding of the underlying principles.

Summary of Solutions

Equation	Solution
$n - 13 = -12$	n = 1
$n + 15 = -10$	n = -25
$\frac{n}{5} = -\frac{1}{5}$	n = -1
$-5n = 1$	$n = -\frac{1}{5}$

In summary, we have solved four linear equations by applying the properties of equality and using inverse operations to isolate the variable. Each equation required a different approach, but the underlying principles remained consistent. By mastering these techniques, you can confidently solve a wide range of linear equations and build a strong foundation in algebra.