Solve The Following Linear Equations: 8. Solve: 8x - 4 + 3x = 7x + X + 14 9. Solve: 8x + 9 - 12x = 4x - 13 - 5x 10. Solve: 5y + 6y - 81 = 7y + 102 + 65y 11. Solve: 16 + 7x - 5 + X = 11x - 3

Solving algebraic equations is a fundamental skill in mathematics. In this section, we will walk through the step-by-step process of solving the equation 8x - 4 + 3x = 7x + x + 14. The primary goal is to isolate the variable x on one side of the equation to determine its value. Let’s delve into each step to ensure a clear understanding.

First, we need to simplify both sides of the equation by combining like terms. On the left-hand side (LHS), we have 8x and 3x, which can be combined to give 11x. So, the LHS simplifies to 11x - 4. On the right-hand side (RHS), we have 7x and x, which combine to give 8x. Thus, the RHS simplifies to 8x + 14. The equation now looks like this: 11x - 4 = 8x + 14. Simplifying both sides is a crucial step because it reduces the complexity of the equation, making it easier to manipulate and solve. By combining like terms, we consolidate the variables and constants, which sets the stage for the subsequent steps in isolating x.

Next, our goal is to isolate the variable term on one side of the equation. To achieve this, we need to eliminate the 8x term from the right-hand side. We can do this by subtracting 8x from both sides of the equation. This maintains the balance of the equation, a fundamental principle in algebra. Subtracting 8x from both sides, we get: 11x - 4 - 8x = 8x + 14 - 8x. This simplifies to 3x - 4 = 14. Isolating the variable term is essential because it brings us closer to solving for x. By performing the same operation on both sides, we ensure that the equation remains balanced, and the solution remains valid.

Now, we need to isolate the constant term on the other side of the equation. Currently, we have 3x - 4 = 14. To eliminate the -4 from the left-hand side, we add 4 to both sides of the equation. This gives us 3x - 4 + 4 = 14 + 4, which simplifies to 3x = 18. By adding 4 to both sides, we ensure that the equation remains balanced and we move closer to isolating x. This step is crucial in the process of solving for x because it separates the variable term from the constant terms.

Finally, to solve for x, we need to divide both sides of the equation by the coefficient of x, which is 3. So, we have 3x = 18. Dividing both sides by 3, we get 3x / 3 = 18 / 3, which simplifies to x = 6. Thus, the solution to the equation 8x - 4 + 3x = 7x + x + 14 is x = 6. This final step completes the process of solving for x. By dividing both sides by the coefficient of x, we isolate x and determine its value. This solution can be verified by substituting x = 6 back into the original equation to ensure that both sides are equal.

In this section, we aim to solve another linear equation: 8x + 9 - 12x = 4x - 13 - 5x. Following a similar approach as before, our goal is to isolate the variable x and find its value. This involves simplifying both sides of the equation, combining like terms, and performing algebraic operations to maintain balance.

The initial step is to simplify both sides of the equation. On the left-hand side (LHS), we have 8x and -12x, which combine to give -4x. So, the LHS simplifies to -4x + 9. On the right-hand side (RHS), we have 4x and -5x, which combine to give -x. Thus, the RHS simplifies to -x - 13. The equation now looks like this: -4x + 9 = -x - 13. Simplifying both sides is a critical step in solving any equation. It reduces the complexity by combining similar terms, which makes the equation easier to manipulate and solve. In this case, by combining the x terms on both sides, we are setting the stage for isolating x.

Next, we need to move the variable terms to one side of the equation. To do this, we can add 4x to both sides of the equation. This eliminates the -4x term from the left-hand side. Adding 4x to both sides, we get: -4x + 9 + 4x = -x - 13 + 4x. This simplifies to 9 = 3x - 13. Moving variable terms to one side is a strategic step that helps in isolating x. By adding 4x to both sides, we ensure that all x terms are on the right-hand side, making it easier to isolate x in the subsequent steps. Maintaining the balance of the equation by performing the same operation on both sides is a fundamental principle in algebra.

Now, we want to isolate the variable term. To achieve this, we need to eliminate the constant term on the right-hand side. We have the equation 9 = 3x - 13. To eliminate the -13, we add 13 to both sides of the equation. This gives us 9 + 13 = 3x - 13 + 13, which simplifies to 22 = 3x. Isolating the variable term is crucial in solving for x. By adding 13 to both sides, we separate the variable term from the constant terms, bringing us closer to the solution. This step ensures that we have the variable term alone on one side, ready for the final step of solving for x.

To solve for x, we need to divide both sides of the equation by the coefficient of x, which is 3. So, we have 22 = 3x. Dividing both sides by 3, we get 22 / 3 = 3x / 3, which simplifies to x = 22/3. Thus, the solution to the equation 8x + 9 - 12x = 4x - 13 - 5x is x = 22/3. This final step provides the value of x that satisfies the original equation. By dividing both sides by the coefficient of x, we isolate x and find its exact value. The solution can be a fraction or a decimal, as is the case here. Verifying the solution by substituting it back into the original equation is always a good practice to ensure accuracy.

In this section, we will solve the equation 5y + 6y - 81 = 7y + 102 + 65y for the variable y. The same principles apply here: simplify both sides, isolate the variable terms, and perform algebraic operations to find the value of y. Let’s break down the process step by step to ensure clarity.

The first step is to simplify both sides of the equation. On the left-hand side (LHS), we have 5y and 6y, which combine to give 11y. So, the LHS simplifies to 11y - 81. On the right-hand side (RHS), we have 7y and 65y, which combine to give 72y. Thus, the RHS simplifies to 72y + 102. The equation now looks like this: 11y - 81 = 72y + 102. Simplifying both sides is a fundamental step in solving equations. By combining like terms, we reduce the complexity of the equation, making it easier to manipulate. This step sets the stage for the subsequent steps of isolating the variable y.

Next, we need to move the variable terms to one side of the equation. To do this, we can subtract 11y from both sides of the equation. This eliminates the 11y term from the left-hand side. Subtracting 11y from both sides, we get: 11y - 81 - 11y = 72y + 102 - 11y. This simplifies to -81 = 61y + 102. Moving variable terms to one side is a strategic step in isolating the variable. By subtracting 11y from both sides, we consolidate the variable terms on the right-hand side, making it easier to isolate y in the following steps. Maintaining the balance of the equation is crucial, and performing the same operation on both sides ensures that the solution remains valid.

Now, we want to isolate the variable term. To achieve this, we need to eliminate the constant term on the right-hand side. We have the equation -81 = 61y + 102. To eliminate the +102, we subtract 102 from both sides of the equation. This gives us -81 - 102 = 61y + 102 - 102, which simplifies to -183 = 61y. Isolating the variable term is a critical step in solving for y. By subtracting 102 from both sides, we separate the variable term from the constant terms, bringing us closer to the solution. This step sets up the final isolation of y.

To solve for y, we need to divide both sides of the equation by the coefficient of y, which is 61. So, we have -183 = 61y. Dividing both sides by 61, we get -183 / 61 = 61y / 61, which simplifies to y = -3. Thus, the solution to the equation 5y + 6y - 81 = 7y + 102 + 65y is y = -3. This final step provides the value of y that satisfies the original equation. By dividing both sides by the coefficient of y, we isolate y and determine its exact value. The solution can be an integer, as is the case here. Verifying the solution by substituting it back into the original equation is a good practice to ensure accuracy.

In this final section, we will solve the equation 16 + 7x - 5 + x = 11x - 3 for the variable x. Following the same systematic approach, we will simplify both sides, combine like terms, and perform algebraic operations to isolate x and find its value. This step-by-step process ensures a clear understanding of how to solve linear equations.

The initial step is to simplify both sides of the equation. On the left-hand side (LHS), we have the constant terms 16 and -5, which combine to give 11. We also have the variable terms 7x and x, which combine to give 8x. So, the LHS simplifies to 8x + 11. On the right-hand side (RHS), we have 11x - 3, which is already in its simplest form. The equation now looks like this: 8x + 11 = 11x - 3. Simplifying both sides is a crucial step in solving equations. It reduces the complexity of the equation by combining like terms, making it easier to manipulate and solve. This step sets the stage for isolating the variable x.

Next, we need to move the variable terms to one side of the equation. To do this, we can subtract 8x from both sides of the equation. This eliminates the 8x term from the left-hand side. Subtracting 8x from both sides, we get: 8x + 11 - 8x = 11x - 3 - 8x. This simplifies to 11 = 3x - 3. Moving variable terms to one side is a strategic step in isolating the variable. By subtracting 8x from both sides, we consolidate the variable terms on the right-hand side, making it easier to isolate x in the following steps. Maintaining the balance of the equation is crucial, and performing the same operation on both sides ensures that the solution remains valid.

Now, we want to isolate the variable term. To achieve this, we need to eliminate the constant term on the right-hand side. We have the equation 11 = 3x - 3. To eliminate the -3, we add 3 to both sides of the equation. This gives us 11 + 3 = 3x - 3 + 3, which simplifies to 14 = 3x. Isolating the variable term is a critical step in solving for x. By adding 3 to both sides, we separate the variable term from the constant terms, bringing us closer to the solution. This step sets up the final isolation of x.

To solve for x, we need to divide both sides of the equation by the coefficient of x, which is 3. So, we have 14 = 3x. Dividing both sides by 3, we get 14 / 3 = 3x / 3, which simplifies to x = 14/3. Thus, the solution to the equation 16 + 7x - 5 + x = 11x - 3 is x = 14/3. This final step provides the value of x that satisfies the original equation. By dividing both sides by the coefficient of x, we isolate x and determine its exact value. The solution can be a fraction or a decimal. As always, verifying the solution by substituting it back into the original equation ensures accuracy.

In conclusion, solving linear equations involves simplifying both sides, moving variable terms to one side, isolating the variable term, and finally, solving for the variable by dividing by its coefficient. Each step is crucial in the process, and maintaining balance by performing the same operations on both sides ensures a correct solution.