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.
Format: Review the step-by-step Example, then test your understanding with the Matched Exercise below it.
Example 1: 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:
\[ (-8a - 5a) + (-10b - b + b - 2b) + (-8 + 4) \]Final Answer: \( -13a - 12b - 4 \)
Solution:
Distribute the terms:
\[ 2a - 16b - 5 + b + b + 6b - 9 - a \]Group like terms:
\[ (2a - a) + (-16b + b + b + 6b) + (-5 - 9) \]Answer: \( a - 8b - 14 \)
Example 2: Solve the equation:
\[ 2(-3x - 5) - (8 - x) = -2(2x + 4) + 12 \]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 \(-1\):
Final Answer: \( x = -22 \)
Check: Substituting \(x = -22\) satisfies both sides of the equation.
Solution:
Distribute and expand:
\[ -2x - 10 + 6 - x = -6x - 12 + 12 \]Combine like terms:
\[ -3x - 4 = -6x \]Add \(6x\) and \(4\) to both sides:
\[ 3x = 4 \]Answer: \( x = \frac{4}{3} \)
Example 3: If \(x > -2\), simplify:
\[ 2|x + 2| - 3x - (-2 - x) + |6 - 9| \]Solution:
Since \(x > -2\), we have \(x + 2 > 0\). According to the absolute value definition:
\[ |x + 2| = x + 2 \]Also evaluate the constant absolute value:
\[ |6 - 9| = |-3| = 3 \]Rewrite the expression and expand:
\[ 2(x + 2) - 3x - (-2 - x) + 3 \] \[ 2x + 4 - 3x + 2 + x + 3 \]Group like terms:
\[ (2x - 3x + x) + (4 + 2 + 3) \]Final Answer: \( 9 \)
Solution:
Since \(x > 3\), we know \(x - 3 > 0\), so \( |x - 3| = x - 3 \).
Also, \( |9 - 20| = |-11| = 11 \).
Substitute and expand:
\[ 2(x - 3) + 6x - 2 + 3x + 11 \] \[ 2x - 6 + 6x - 2 + 3x + 11 \]Group like terms:
Answer: \( 11x + 3 \)
Example 4: Find the slope and y-intercept of the line:
\[ 2y - 3x = 10 \]Solution:
Rewrite the equation in slope-intercept form (\(y = mx + b\)):
\[ 2y = 3x + 10 \]Divide everything by 2:
\[ y = \frac{3}{2}x + 5 \]Final Answer: The slope is \( m = \frac{3}{2} \) and the y-intercept is \( (0, 5) \).
Solution:
Rewrite in slope-intercept form:
\[ -3y = 6x + 7 \]Divide by -3:
\[ y = -2x - \frac{7}{3} \]Answer: The slope is \( -2 \) and the y-intercept is \( (0, -\frac{7}{3}) \).
Example 5: Find the equation of the line passing through \((2, 3)\) and \((4, 1)\).
Solution:
First, compute the slope (\(m\)):
\[ m = \frac{1 - 3}{4 - 2} = \frac{-2}{2} = -1 \]Use the point-slope form \(y - y_1 = m(x - x_1)\):
\[ y - 3 = -1(x - 2) \]Rewrite in slope-intercept form:
\[ y - 3 = -x + 2 \]Final Answer: \( y = -x + 5 \)
Solution:
First, compute the slope:
\[ m = \frac{1 - 3}{-1 - 0} = \frac{-2}{-1} = 2 \]Since the point \((0, 3)\) is given, we already know the y-intercept is \(b = 3\).
Using \(y = mx + b\):
Answer: \( y = 2x + 3 \)