Algebra Tutorial with Detailed Solutions

This tutorial provides clear explanations and step-by-step solutions to common algebra topics: simplifying expressions, solving linear equations (including absolute value equations), finding slopes and intercepts, and writing equations of lines. Matched practice exercises are included for reinforcement.

Try a self-test on algebra problems covering similar topics.

Example 1: Simplifying an Algebraic Expression

Simplify the expression:

\[ 2(-4a - 5b) - (8 + b) + b + (-2b + 4) - 5a \]

Solution

First, distribute all multiplications:

\[ -8a - 10b - 8 - b + b - 2b + 4 - 5a \]

Next, group like terms:

\[ (-8a - 5a) + (-10b - 2b) + (-8 + 4) \] \[ -13a - 12b - 4 \]

Final Answer: \[ -13a - 12b - 4 \]

Matched Exercise 1

Simplify:

\[ 2(a - 8b) - (5 - b) + b + (6b - 9) - a \]

View Solution

Example 2: Solving a Linear Equation

Solve:

\[ 2(-3x - 5) - (8 - x) = -2(2x + 4) + 12 \]

Solution

Distribute on both sides:

\[ -6x - 10 - 8 + x = -4x - 8 + 12 \]

Combine like terms:

\[ -5x - 18 = -4x + 4 \]

Add 18 to both sides:

\[ -5x = -4x + 22 \]

Add \(4x\) to both sides:

\[ -x = 22 \]

Multiply both sides by \(-1\):

\[ x = -22 \]

Check: Substituting \(x = -22\) satisfies both sides of the equation.

Final Answer: \[ x = -22 \]

Matched Exercise 2

\[ 2(-x - 5) - (-6 + x) = -3(2x + 4) + 12 \]

Example 3: Simplifying an Expression with Absolute Value

If \(x > -2\), simplify:

\[ 2|x + 2| - 3x - (-2 - x) + |6 - 9| \]

Solution

Since \(x > -2\), we have \(x + 2 > 0\), so:

\[ |x + 2| = x + 2 \]

Also:

\[ |6 - 9| = |-3| = 3 \]

Rewrite the expression:

\[ 2(x + 2) - 3x - (-2 - x) + 3 \]

Expand and simplify:

\[ 2x + 4 - 3x + 2 + x + 3 \] \[ (2x - 3x + x) + (4 + 2 + 3) = 9 \]

Final Answer: \[ 9 \]

Matched Exercise 3

If \(x > 3\), simplify:

\[ 2|x - 3| + 6x - (2 - 3x) + |9 - 20| \]

View Solution

Example 4: Finding the Slope and Y-Intercept

Find the slope and y-intercept of the line:

\[ 2y - 3x = 10 \]

Solution

Rewrite the equation in slope-intercept form \(y = mx + b\):

\[ 2y = 3x + 10 \] \[ y = \frac{3}{2}x + 5 \]

The slope is \(m = \frac{3}{2}\) and the y-intercept is \((0, 5)\).

Matched Exercise 4

\[ -3y - 6x = 7 \]

View Solution

Example 5: Equation of a Line Through Two Points

Find the equation of the line passing through \((2, 3)\) and \((4, 1)\).

Solution

First, compute the slope:

\[ m = \frac{1 - 3}{4 - 2} = -1 \]

Use the point-slope form:

\[ y - 3 = -1(x - 2) \]

Rewrite in slope-intercept form:

\[ y = -x + 5 \]

Final Answer: \[ y = -x + 5 \]

Matched Exercise 5

Find the equation of the line through \((0, 3)\) and \((-1, 1)\).

View Solution

More Algebra Resources

Algebra Problems
Interactive Tutorial on Slopes
Interactive Tutorial on Lines
More Algebra Problems with Solutions