Question

solve the following linear equations for the unknown variable.
note: all necessary algebraic operations required to isolate the vari-
able must be shown to receive full credit.
(a) $x + 19 = 34$ (10 points)
(b) $\frac{2y - 3}{4} = 15$ (10 points)
(c) $-3z + 25 = -8$ (10 points)
(d) $23x + 45y - 31z = 10$ for $x = -2$ and $y = 1$ (10 points)

Explanation:

Part (a)

Step1: Subtract 19 from both sides

To isolate \( x \), we subtract 19 from both sides of the equation \( x + 19 = 34 \).
\( x + 19 - 19 = 34 - 19 \)

Step2: Simplify both sides

Simplifying the left side gives \( x \), and simplifying the right side: \( 34 - 19 = 15 \).
\( x = 15 \)

Step1: Multiply both sides by 4

To eliminate the denominator, we multiply both sides of the equation \( \frac{2y - 3}{4} = 15 \) by 4.
\( 4\times\frac{2y - 3}{4}=15\times4 \)

Step2: Simplify both sides

Simplifying the left side gives \( 2y - 3 \), and the right side: \( 15\times4 = 60 \). So we have \( 2y - 3 = 60 \).

Step3: Add 3 to both sides

Add 3 to both sides to isolate the term with \( y \): \( 2y - 3 + 3 = 60 + 3 \)

Step4: Simplify both sides

Simplifying the left side gives \( 2y \), and the right side: \( 60 + 3 = 63 \). So \( 2y = 63 \).

Step5: Divide both sides by 2

Divide both sides by 2 to solve for \( y \): \( y=\frac{63}{2}= 31.5 \)

Step1: Subtract 25 from both sides

To isolate the term with \( z \), we subtract 25 from both sides of the equation \( - 3z + 25=-8 \).
\( -3z + 25 - 25=-8 - 25 \)

Step2: Simplify both sides

Simplifying the left side gives \( - 3z \), and the right side: \( -8 - 25=-33 \). So \( -3z=-33 \).

Step3: Divide both sides by - 3

Divide both sides by - 3 to solve for \( z \): \( \frac{-3z}{-3}=\frac{-33}{-3} \)

Step4: Simplify both sides

Simplifying gives \( z = 11 \).

Answer:

\( x = 15 \)

Part (b)