Solve For { T $} . . . { -35 = 3w + 4(t + 7) \} Simplify Your Answer As Much As Possible.

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of linear equation, which involves isolating the variable $t$. We will use a step-by-step approach to simplify the equation and find the value of $t$.

The Given Equation

The given equation is:

$$-35 = 3w + 4(t + 7)$$

Our goal is to isolate the variable $t$ and simplify the equation as much as possible.

Step 1: Distribute the 4

To start solving the equation, we need to distribute the 4 to the terms inside the parentheses:

$$-35 = 3w + 4t + 28$$

Step 2: Combine Like Terms

Next, we need to combine the like terms on the right-hand side of the equation:

$$-35 = 3w + 4t + 28$$

Since there are no like terms on the left-hand side, we can move on to the next step.

Step 3: Isolate the Variable $t$

To isolate the variable $t$, we need to get rid of the terms involving $w$. We can do this by subtracting $3w$ from both sides of the equation:

$$-35 - 3w = 4t + 28$$

Step 4: Simplify the Equation

Now, we need to simplify the equation by combining the constants on the left-hand side:

$$-35 - 3w = 4t + 28$$

$$-35 - 3w - 28 = 4t$$

$$-63 - 3w = 4t$$

Step 5: Solve for $t$

Finally, we can solve for $t$ by dividing both sides of the equation by 4:

$$\frac{-63 - 3w}{4} = t$$

Conclusion

In this article, we have solved a linear equation involving the variable $t$. We used a step-by-step approach to simplify the equation and isolate the variable $t$. The final solution is:

$$\frac{-63 - 3w}{4} = t$$

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS).
• Use parentheses to group terms and make it easier to simplify the equation.
• Combine like terms to simplify the equation and make it easier to solve.
• Isolate the variable by getting rid of the terms involving other variables.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future trends.

Common Mistakes

• Failing to follow the order of operations (PEMDAS).
• Not using parentheses to group terms.
• Not combining like terms.
• Not isolating the variable.

Conclusion

Introduction

In our previous article, we explored the step-by-step process of solving linear equations. However, we understand that sometimes, it's not just about following a set of rules, but also about understanding the underlying concepts and addressing common questions and concerns. In this article, we will provide a Q&A guide to help you better understand and tackle linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it's an equation that can be written in the form:

ax + by = c

where a, b, and c are constants, and x and y are variables.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, whereas a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example:

Linear equation: 2x + 3y = 5

Quadratic equation: x^2 + 4x + 4 = 0

Q: How do I solve a linear equation with multiple variables?

A: To solve a linear equation with multiple variables, you need to isolate one variable by getting rid of the other variables. You can do this by using the following steps:

1. Use the distributive property to expand the equation.
2. Combine like terms.
3. Isolate one variable by getting rid of the other variables.

For example:

2x + 3y = 5

x + 2y = 3

To solve for x, you can isolate x by getting rid of y:

2x + 3(3 - x) = 5

2x + 9 - 3x = 5

-x = -4

x = 4

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation with one or more variables, whereas a system of linear equations is a set of two or more linear equations with the same variables. For example:

Linear equation: 2x + 3y = 5

System of linear equations:

2x + 3y = 5

x - 2y = -3

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to find the values of the variables that satisfy all the equations. You can do this by using the following methods:

1. Substitution method: Substitute the expression for one variable from one equation into the other equation.
2. Elimination method: Add or subtract the equations to eliminate one variable.
3. Graphical method: Graph the equations on a coordinate plane and find the point of intersection.

For example:

2x + 3y = 5

x - 2y = -3

You can use the substitution method to solve for x:

x = -3 + 2y

Substitute this expression for x into the first equation:

2(-3 + 2y) + 3y = 5

-6 + 4y + 3y = 5

7y = 11

y = 11/7

Now that you have the value of y, you can substitute it into one of the original equations to find the value of x:

x - 2(11/7) = -3

x - 22/7 = -3

x = -3 + 22/7

x = (-21 + 22)/7

x = 1/7

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

• Failing to follow the order of operations (PEMDAS).
• Not using parentheses to group terms.
• Not combining like terms.
• Not isolating the variable.
• Not checking the solution by plugging it back into the original equation.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By understanding the underlying concepts and addressing common questions and concerns, you can become more confident and proficient in solving linear equations. Remember to follow the order of operations, use parentheses, combine like terms, and isolate the variable to ensure accurate solutions.