91Ó°ÊÓ

To solve for the variable \(y\) in a linear equation, you need to isolate \(y\) on one side of the equation. This involves basic algebraic operations such as addition, subtraction, multiplication, or division.
In the given problem, we have the linear equation: \(y = 4 - 3x\). To find \(y\), substitute the value of \(x\) into this equation.

For \(x = -2\):
Substitute \( -2 \) into the equation: \[y = 4 - 3(-2)\] Simplify the multiplication operation: \[y = 4 + 6 \] Add the numbers: \[ y = 10\]
So, when \(x = -2\), \(y = 10\).

For \(x = 4\):
Substitute \ 4 \ into the equation: \[y = 4 - 3(4) \] Simplify the multiplication operation: \[y = 4 - 12 \] Subtract the numbers: \[ y = -8 \]
Therefore, when \(x = 4\), \(y = -8 \).

The key to solving for \(y\) is to carefully perform each arithmetic step, ensuring that all operations are correctly simplified.

The substitution method is a technique used to find solutions to algebraic equations by substituting values. In our problem, we use substitution to replace the variable \(x\) with specific values to find \(y\).

Here's how the substitution method is applied:
Start with the linear equation given: \(y = 4 - 3x\) Identify the value of \(x\) that you need to substitute.

• For \(x = -2\), substitute \( -2 \) into the equation and solve.
• For \(x = 4\), substitute \( 4 \) into the equation and solve.

The substitution method simplifies the process of solving equations, making it systematic and straightforward. When you substitute the value into the equation, follow the order of operations: perform multiplication first, then addition or subtraction.

This method is particularly useful when dealing with single variable equations like linear functions, as it provides a clear path to the solution.

Linear functions are algebraic equations that represent straight lines on a graph. They have the general form: \[ y = mx + b \] where:

• \(y\) is the dependent variable, often representing the output.
• \(x\) is the independent variable, often representing the input.
• \(m\) is the slope of the line, showing the rate of change.
• \(b\) is the \(y\)-intercept, indicating where the line crosses the \(y\)-axis.

In our problem, the equation \(y = 4 - 3x\) can be seen in this form. Here:

• \( m = -3\), the coefficient of \(x\), indicates the slope.
• \( b = 4\), the constant term, shows the \(y\)-intercept.

This means the line will cross the \(y\)-axis at \(4\) and will slope downwards because the slope is negative.

Linear functions are foundational in algebra as they model relationships with constant rates of change. Understanding them helps in graphing and analyzing trends and behaviors in various fields, from economics to engineering.