Problem 16 - College Algebra Essentials

Question

Solve each system. $$\left\\{\begin{array}{l} x+y=4 \\ x+z=4 \\ y+z=4 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is x = 2, y = 2, z = 2.

Step 1: Identification of substitute equations

Notice that we can isolate z in the second equation and y in the first. This makes it easier to substitute into the third equation. So, \( x = 4 - z\) from the second equation and \( y = 4 - x\) from the first equation.

Step 2: Substitution into the third equation

Now we substitute these expressions into the third equation:\( 4 - x = 4 - z\)which simplifies to\( x = z\).

Step 3: Solve for the remaining variables

Next, we substitute \( x = z\), into the second equation:\( z + y = 4\)which simplifies to\( y = 4 - z\)and \( y = 4 - x\)which means \( y = 4 - x = 4 - z = 2\). Now substitute \( y = 2\), into the first equation:\( x + 2 = 4\)which simplifies to\( x = 2\). Since \( x = z\), it follows that \( z = 2\).