Problem 36 - Elementary Algebra

Question

For the following problems, solve the linear equations in two variables. $$ 3(t+2)=4(s-9), \text { if } s=1 $$

Step-by-Step Solution

Answer

Answer: The value of t is -38/3.

Step 1: Substitute the value of s in the equation

We are given that s = 1. Let's substitute this value in the equation. $$ 3(t+2)=4(1-9) $$

Step 2: Simplify the equation

Now, let's simplify the equation by simplifying the expressions inside the parentheses and then expanding. $$ 3(t+2)=4(-8) $$

Step 3: Expand the equation

Now, let's expand the equation by multiplying both sides by the respective constant factors. $$ 3t + 6 = -32 $$

Step 4: Solve for t

Finally, let's solve the equation for t by isolating t on one side. $$ 3t = -32 - 6 $$ $$ 3t = -38 $$ Now, divide both sides by 3: $$ t = -\frac{38}{3} $$ So, the solution is: $$t = -\frac{38}{3}$$.

Key Concepts

Understanding Variables in Linear Equations

In linear equations, variables are symbols that stand in for unknown numbers. They help in forming equations by representing these unknown quantities. In the exercise given, the variables are \( t \) and \( s \). Here, \( t \) is the variable for which we need to solve, while \( s \) is a given value.

Variables are crucial because they allow equations to express relationships between numbers that are not yet known. They make it possible to use algebraic methods to solve problems. Keep in mind:

• Variables can represent different values.
• Equations are balanced scales, meaning the operations you do to one side are also done to the other.

This balance is what allows us to solve for the unknowns.

Substitution: Replacing Variables with Values

Substitution is a simple concept where we replace a variable with a given or known value. This simplifies the equation, making it straightforward to solve. In the exercise, we were given \( s = 1 \). This means wherever \( s \) appears in the equation, we substitute it with \( 1 \).

Here’s how it looks: from the original equation \( 3(t+2)=4(s-9) \), after substitution, it becomes \( 3(t+2)=4(1-9) \). This step is often the starting point in solving equations, especially when one variable's value is provided. Remember that:

• Substitution changes the equation to only include one unknown.
• Conduct arithmetic operations carefully after substitution.

This helps in uncovering one variable's value.

Simplification: Making the Equation Manageable

Simplification is the art of reducing complexity in an equation, allowing you to easily work through it. It involves performing basic arithmetic operations and combining like terms. After substitution, our focus was on simplifying \( 3(t + 2) = 4(-8) \).

This was achieved by calculating inside the parentheses and multiplying where possible, leading to \( 3(t + 2) = -32 \). Simplification can also involve:

• Combining like terms, which are terms that have the same variables raised to the same power.
• Performing operations inside parentheses first, according to the order of operations.

This makes an equation easier to handle, preparing it for the final step of isolating the desired variable.

Solving Equations: Finding the Unknown

Solving the equation involves isolating the variable you are interested in. From the simplified equation \( 3t + 6 = -32 \), we aimed to isolate \( t \) on one side. This was done by reversing operations using addition or subtraction, and finally division.

Step by step, it looked like this:

• Subtract 6 from both sides: \( 3t = -32 - 6 \), resulting in \( 3t = -38 \).
• Divide each side by 3 to solve for \( t \): \( t = -\frac{38}{3} \).

Achieving this requires systematic removal of anything added or multiplied to \( t \). Think of solving equations as similar to peeling an onion, layer by layer, until you reach the core—the value of the unknown. Remember, always maintain equation balance!