91Ó°ÊÓ

The **substitution method** is a popular algebraic technique for solving systems of linear equations. This method involves solving one of the equations for one variable and then substituting this expression into another equation.

In the given exercise, we used this method effectively:

1. We started by solving the third equation, which was the simplest, for x. This gave us x in terms of a numerical value.
2. Next, we substituted this x value into the first equation to solve for y.
3. Finally, we substituted the found y value into the second equation to solve for z.

By breaking down the process into steps, the substitution method simplifies the system of equations, making them more manageable.

A **linear equation** represents a straight line and has the general form: ax + by + cz + ... = d, where a, b, c, and d are constants and x, y, z, etc., are variables.

In our exercise, we were given the following linear equations:

• \[ x - y = 3 \]
• \[ y + z = 8 \]
• \[ 2x = 7 \]

These equations are defined by relationships between the variables with respect to constants and among themselves.

By solving these equations, we were able to fine the values of x, y, and z that satisfy all three equations simultaneously. Linear equations are a fundamental concept in algebra essential for understanding more complex mathematical concepts.

The process of **solving for variables** involves isolating a variable on one side of the equation.

In our exercise:

1. We solved for x by isolating it in the third equation:\[ 2x = 7 \]by dividing both sides by 2, giving us:\[ x = \frac{7}{2} \].
2. We then used this x value and substituted it into the first equation to isolate and solve for y:\[ \frac{7}{2} - y = 3 \]and we reorganized to solve for y:\[ y = \frac{1}{2} \].
3. Finally, with y known, we substituted it into the second equation to solve for z:\[ \frac{1}{2} + z = 8 \]and reorganized to find z:\[ z = \frac{15}{2} \].

By carefully manipulating equations, we found the values for all variables. This logical sequence of steps ensures that each variable is accurately determined based on the others.