91Ó°ÊÓ

When solving linear equations, one of the primary goals is to isolate the variable you are solving for, in this case, the variable \( y \). This process involves rearranging the equation so that \( y \) is on one side by itself. To begin isolating \( y \) in an equation like \( 7x - 3y = 22 \), you need to perform operations that will remove the term with \( x \) from the side with \( y \).

• Subtract \( 7x \) from both sides: This operation cancels out the \( x \) term on the left side and brings it over to the right side, resulting in \( -3y = -7x + 22 \).
• At this point, \( y \) is still multiplied by -3, so you will divide the entire equation by -3 to solve for \( y \).

Breaking down the equation into smaller, manageable steps helps make the math process straightforward and error-free. Isolating the variable is key to understanding how different components of the equation interact.

Algebraic manipulation is a critical skill for solving equations and finding the values of unknowns. It involves various operations, such as addition, subtraction, multiplication, and division, to transform and simplify equations. For example, when presented with an equation like \( 5x + 4y = -12 \), algebraic manipulation allows you to isolate \( y \) and solve the equation in stages:

• Beginning with subtraction, you eliminate the \( 5x \) term on the left side by subtracting it from both sides, which gives \( 4y = -5x - 12 \).
• Then, by dividing each term by 4, you solve for \( y \): \( y = -\frac{5}{4}x - 3 \).

These steps demonstrate how algebraic manipulation simplifies equations and helps isolate the variables, making it possible to find solutions in a logical flow. Remember that these manipulations should maintain the equality of both sides, ensuring the equation remains balanced.

Handling equations with two variables, like the ones given, means you are dealing with a system of linear equations. Each equation represents a line on a graph. When you solve for one variable in terms of the other, you express the equation as a specific linear function of that variable. Consider the solved form of \( y \) in the equations \( y = \frac{7}{3}x - \frac{22}{3} \) and \( y = -\frac{5}{4}x - 3 \). Each equation indicates the slope and intercept of the lines they represent:

• The equation \( y = \frac{7}{3}x - \frac{22}{3} \) describes a line with a slope of \( \frac{7}{3} \) and a y-intercept of \( -\frac{22}{3} \).
• The equation \( y = -\frac{5}{4}x - 3 \) has a slope of \( -\frac{5}{4} \) and a y-intercept of \( -3 \).

Solving two-variable equations helps understand the relationship between two changing quantities represented by \( x \) and \( y \). This understanding is crucial for more complex algebraic studies and helps lay the groundwork for analyzing systems of equations geometrically and algebraically.