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A linear equation is an algebraic statement where each term is either a constant or the product of a constant with a single variable. It represents a straight line when graphed on a coordinate plane. These equations have a standard form:

• One or more variables
• Constant coefficients
• No products of variables (e.g., no exponents)
• No variable inside a function like a square root or trigonometric function

A typical linear equation with two variables is written as: \( y = mx + b \).
Here, \( m \) is the slope of the line, detailing how steep the line is, and \( b \) is the y-intercept, revealing where the line crosses the y-axis.
Linear equations are fundamental in math and can solve problems related to real-world phenomena, such as calculating distances or predicting costs.
In our example, our linear equation is in slope-intercept form, \( y = \frac{1}{2}x - 3 \), showing the slope as \( \frac{1}{2} \) and the y-intercept at -3.

The substitution method is a way of solving equations by replacing one variable with an equivalent expression. This is especially helpful for solving systems of equations, but it's also useful for single equations when you have a specific value for a variable.
Here’s a quick run-through of the substitution process:

• Identify the equation and the value given for a variable.
• Replace the variable in the equation with the given value. This step is called substitution.
• Simplify the new equation to solve for the remaining variable.

In our problem, we substituted \( x = -4 \) into the equation \( y = \frac{1}{2}x - 3 \). By calculating \( \frac{1}{2} \times -4 \), we transform the equation and all terms involving \( x \) into numbers. This makes solving for another variable, like \( y \), straightforward.

Once you substitute the value of \( x \) into the equation, your task is to solve for \( y \). Solving for a variable means isolating it on one side of the equation to find its value.
Follow these steps to solve for \( y \):

• First, perform any necessary arithmetic operations from the substitution.
• In our example, we calculated \( \frac{1}{2} \times -4 = -2 \).
• Use the results from the arithmetic operations to continue simplifying the equation.
• Replace the expression that contained \( x \) and solve for \( y \).
• For instance, \( y = -2 - 3 \) simplifies to \( y = -5 \).

Thus, solving for \( y \), you confirm its final value, completing the equation with clarity and precision. By practicing this process, you'll become adept at maneuvering through equations with confidence and ease.