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When working with algebraic equations, one of the primary tasks is solving for a variable. This means isolating the variable of interest on one side of the equation. To accomplish this, we typically use basic arithmetic operations such as addition, subtraction, multiplication, or division.
For instance, consider the equation presented in the exercise: \(3x + y = 4\). Here, we need to isolate the variable \(y\). We do this by eliminating \(3x\) from the left side to have \(y\) standing alone.

• Subtract \(3x\) from both sides:
• This action helps eliminate \(3x\) from the left side, leaving \(y\) by itself.

Thus, the equation becomes \(y = 4 - 3x\), effectively solving for \(y\) in terms of \(x\). This process allows us to express one variable with respect to another, crucial for understanding relationships between variables in equations.

Equations that involve two variables, like the one in the exercise, represent a relationship between those two variables. In the given equation \(3x + y = 4\), both \(x\) and \(y\) conform to a specific mathematical relationship.
It's important to note that in such equations, any value of \(x\) corresponds to a value of \(y\) when you solve for \(y\) and vice versa.

• By re-arranging the equation to solve for \(y\), every instance of \(x\) will give a specific \(y\).
• This ability to pivot between variables lets us explore various scenarios where the equation holds true.

Equations with two variables find applications in multiple areas, such as finding cost-price relationships, analyzing trends, and solving systems of equations in mathematics.

A key feature of the given equation is that it is linear. Linear equations are represented in the standard form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. For our example, \(A = 3\), \(B = 1\), and \(C = 4\).
Linear equations graph as straight lines, which makes them easy to plot and analyze. These lines represent constant rates of change between variables.

• In the solution \(y = 4 - 3x\), \(-3\) is the slope, indicating how much \(y\) changes with each unit change in \(x\).
• The number \(4\) is the y-intercept, showing where the line crosses the y-axis.

Understanding linear equations furnishes us with solid tools for predicting trends and making informed decisions based on variable relationships.