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 SAT Math Lesson Plan 1: Linear Equations and Expressions

 

Lesson Summary

This 45-minute lesson is part of a comprehensive 35-lesson SAT Math prep course. It focuses on solving and analyzing linear equations, a key topic within the SAT's Heart of Algebra, which makes up approximately 35% of the SAT Math test. Students will learn to manipulate algebraic expressions, solve for variables, graph linear equations, and interpret slope and intercepts. The lesson includes step-by-step examples, guided practice, and self-study tips to reinforce learning. A review section summarizes key takeaways, and a multiple-choice quiz provides an opportunity for self-assessment. By the end of this lesson, students will have a strong foundation in working with linear equations, a critical skill for success on the SAT.

Lesson Objectives

  • Solve single-variable linear equations.
  • Simplify and manipulate algebraic expressions.
  • Identify and interpret different forms of linear equations (slope-intercept, standard, and point-slope form).
  • Graph linear equations and understand the meaning of slope and y-intercept in different contexts.

Common Core Standards

  • CCSS.MATH.CONTENT.HSA.REI.B.3 – Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  • CCSS.MATH.CONTENT.HSA.SSE.A.1 – Interpret expressions that represent a quantity in terms of its context.
  • CCSS.MATH.CONTENT.HSF.IF.B.4 – Interpret key features of graphs and tables in terms of the situation they represent, particularly the slope and y-intercept in the context of a linear equation.
  • CCSS.MATH.CONTENT.HSF.LE.A.2 – Construct linear functions, including interpreting the parameters in context.

Prerequisite Skills

  • Understanding basic algebraic operations.
  • Ability to combine like terms and apply the distributive property.
  • Familiarity with solving for a variable in simple equations.
  • Basic understanding of graphing points on a coordinate plane.

Key Vocabulary

  • Linear equation – An equation that forms a straight line when graphed.
  • Expression – A mathematical phrase containing numbers, variables, and operations but no equality sign.
  • Coefficient – A number that multiplies a variable (e.g., in 3x, the coefficient is 3).
  • Distributive Property – A rule stating a(b + c) = ab + ac.
  • Slope-intercept form – The equation y = mx + b, where m is the slope and b is the y-intercept.
  • Slope – A measure of the steepness of a line, calculated as the ratio of vertical change to horizontal change.
  • Y-intercept – The point where the line crosses the y-axis.

 

 

Warm-Up

Goal: Activate prior knowledge with a quick review.

Solve for x in the following equations:

  • \( 2x + 5 = 11 \)
    • Subtract 5 from both sides: \( 2x = 6 \)
    • Divide by 2: \( x = 3 \)
  • \( 3(x - 2) = 12 \)
    • Distribute: \( 3x - 6 = 12 \)
    • Add 6 to both sides: \( 3x = 18 \)
    • Divide by 3: \( x = 6 \)

Self-Study Tip: Try solving these in 2-3 minutes before reviewing the full lesson.

 

Teach

In this section, we’ll explore different ways to represent and analyze linear equations. We’ll cover:

  • Solving linear equations using inverse operations.
  • Simplifying algebraic expressions.
  • Converting equations to slope-intercept form.
  • Graphing linear equations using slope and y-intercept.
  • Writing equations using the point-slope form.
  • Recognizing parallel and perpendicular lines.

Understanding these concepts will help you solve SAT questions efficiently and recognize different equation formats.

Example 1: Solving Linear Equations

Problem: Solve for x in \( 5x - 7 = 18 \).

Step 1: Understand the Goal

  • We need to isolate x on one side of the equation.
  • This is done using inverse operations (addition/subtraction and multiplication/division).

Step 2: Add 7 to Both Sides

  • Since the left side has \( -7 \), add \( 7 \) to both sides to cancel it out.
  • \( 5x - 7 + 7 = 18 + 7 \)
  • \( 5x = 25 \)

Step 3: Divide by 5

  • Now, divide both sides by \( 5 \) to solve for \( x \).
  • \( x = \frac{25}{5} \)
  • \( x = 5 \)

Step 4: Check Your Solution

  • Substituting \( x = 5 \) back into the original equation:
  • \( 5(5) - 7 = 25 - 7 = 18 \), which is correct.

Final Answer: \( x = 5 \)

Example 2: Simplifying Expressions

Problem: Simplify \( 3(x + 4) - 2x \).

Step 1: Apply the Distributive Property

  • Distribute \( 3 \) to both terms inside the parentheses:
  • \( 3(x + 4) = 3x + 12 \)

Step 2: Rewrite the Expression

  • Now, replace the original expression with what we found:
  • \( 3x + 12 - 2x \)

Step 3: Combine Like Terms

  • Identify terms with x: \( 3x \) and \( -2x \).
  • Since \( 3x - 2x = x \), we simplify:
  • \( x + 12 \)

Step 4: Verify

  • Check by substituting a value for \( x \), like \( x = 2 \).
  • Original: \( 3(2 + 4) - 2(2) = 3(6) - 4 = 18 - 4 = 14 \)
  • Simplified: \( 2 + 12 = 14 \), which matches.

Final Answer: \( x + 12 \)

Example 3: Converting to Slope-Intercept Form

Problem: Convert the equation \( 2x - y = 4 \) to slope-intercept form.

Step 1: Understand the Goal

  • Slope-intercept form is \( y = mx + b \), where:
  • \( m \) is the slope and \( b \) is the y-intercept.
  • We need to solve for \( y \).

Step 2: Subtract \( 2x \) from Both Sides

  • To isolate \( y \), move \( 2x \) to the right side:
  • \( -y = -2x + 4 \)

Step 3: Multiply by -1

  • Since \( y \) is negative, multiply everything by \( -1 \) to make it positive:
  • \( y = 2x - 4 \)

Step 4: Identify the Slope and Y-Intercept

  • The slope is \( m = 2 \).
  • The y-intercept is \( b = -4 \).

Step 5: Check Your Work

  • Pick a value for \( x \), like \( x = 3 \).
  • Original equation: \( 2(3) - y = 4 \) → \( 6 - y = 4 \) → \( y = 2 \).
  • Converted equation: \( y = 2(3) - 4 = 6 - 4 = 2 \), which matches.

Final Answer: \( y = 2x - 4 \)

Example 4: Graphing a Linear Function

 

Linear functions

 

Problem: Graph the equation \( y = \frac{1}{2}x - 3 \).

Step 1: Identify the Slope and Y-Intercept

  • Slope (m): \( \frac{1}{2} \) (rise over run: up 1, right 2).
  • Y-Intercept (b): -3 (crosses the y-axis at (0,-3)).

Step 2: Plot the Y-Intercept

  • Start by placing a point at (0,-3) on the graph.

Step 3: Use the Slope to Find Another Point

  • From (0,-3), move up 1 unit and right 2 units.
  • Plot another point at (2,-2).

Step 4: Draw the Line

  • Connect the points with a straight line.

Step 5: Verify

  • Pick another value for \( x \), like \( x = 4 \).
  • Calculate \( y = \frac{1}{2}(4) - 3 = 2 - 3 = -1 \).
  • Check that (4,-1) is also on your graph.

Conclusion: The graph is a straight line passing through (-3) on the y-axis with a slope of \( \frac{1}{2} \), meaning it increases gradually from left to right.

Example 5: Writing an Equation Using Point-Slope Form

Problem: Write an equation of the line passing through (4, -2) with a slope of \( \frac{3}{4} \).

Step 1: Use the Point-Slope Formula

  • The point-slope form is: \[ y - y_1 = m(x - x_1) \]

Step 2: Substitute Values

  • \( m = \frac{3}{4} \), \( x_1 = 4 \), \( y_1 = -2 \).
  • Substituting: \[ y - (-2) = \frac{3}{4}(x - 4) \]

Step 3: Simplify

  • \( y + 2 = \frac{3}{4}x - 3 \)
  • Subtract 2: \[ y = \frac{3}{4}x - 5 \]

Conclusion: The equation of the line in slope-intercept form is \( y = \frac{3}{4}x - 5 \).

Example 6: Recognizing Parallel and Perpendicular Lines

 

Parallel lines

 

Problem: Determine whether the lines \( y = 2x + 5 \) and \( y = 2x - 3 \) are parallel, perpendicular, or neither.

Step 1: Identify the Slopes

  • The slope of \( y = 2x + 5 \) is \( m_1 = 2 \).
  • The slope of \( y = 2x - 3 \) is \( m_2 = 2 \).

Step 2: Compare Slopes

  • Since \( m_1 = m_2 \), the lines are parallel.

Bonus: What if the Lines Were Perpendicular?

  • If one line had a slope of \( 2 \), the perpendicular line would have a slope of \( -\frac{1}{2} \) (negative reciprocal).

Conclusion: Two lines are parallel if they have the same slope. Two lines are perpendicular if their slopes are negative reciprocals.

 

 

Review

Key Takeaways

  • Linear equations represent straight-line relationships between variables.
  • To solve linear equations, use inverse operations such as addition, subtraction, multiplication, and division.
  • Simplifying expressions involves combining like terms and using the distributive property.
  • The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
  • Graphing a line requires identifying the y-intercept and using the slope to find additional points.
  • The point-slope form of a line is useful for writing equations when given a point and the slope.
  • Two lines are parallel if they have the same slope.
  • Two lines are perpendicular if their slopes are negative reciprocals.

Example 1: Solving a Linear Equation

Problem: Solve for \( x \) in the equation \( 3x + 7 = 19 \).

Step 1: Subtract 7 from both sides

  • \( 3x + 7 - 7 = 19 - 7 \)
  • \( 3x = 12 \)

Step 2: Divide by 3

  • \( x = \frac{12}{3} \)
  • \( x = 4 \)

Final Answer: \( x = 4 \).

Example 2: Converting to Slope-Intercept Form

Problem: Convert \( 4x - 2y = 8 \) to slope-intercept form.

Step 1: Subtract \( 4x \) from both sides

  • \( -2y = -4x + 8 \)

Step 2: Divide by -2

  • \( y = 2x - 4 \)

Final Answer: \( y = 2x - 4 \).

Example 3: Identifying Parallel and Perpendicular Lines

Problem: Determine whether the lines y = 3x + 1 and y = -1/3x + 4 are parallel, perpendicular, or neither.

Step 1: Identify the Slopes

  • The slope of y = 3x + 1 is m = 3.
  • The slope of y = -1/3x + 4 is m = -1/3.

Step 2: Compare the Slopes

  • Since both slopes are different, the lines are not parallel.
  • The two slopes are negative reciprocals of each other because 3 and -1/3 multiply to -1.

Step 3: Conclusion

  • Because the slopes are negative reciprocals, the lines are perpendicular.

Final Answer: The lines are perpendicular.

Multimedia Resources

See a collection of Media4Math resources that support this topic. Click on this link.

 

 

Quiz

  1. Solve for x: \( 7x - 4 = 24 \)
    A) 2 
    B) 4 
    C) 6 
    D) 8 
     
  2. Solve for x: \( 3(x + 5) = 2x + 15 \)
    A) -3 
    B) 0 
    C) 3 
    D) 5 
     
  3. Simplify: \( 4(2x - 3) + 6x \)
    A) 8x - 3 
    B) 14x - 12 
    C) 10x - 3 
    D) 12x - 6 
     
  4. Convert to slope-intercept form: \( 5x - 3y = 12 \)
    A) \( y = \frac{5}{3}x - 4 \) 
    B) \( y = \frac{3}{5}x + 4 \) 
    C) \( y = \frac{5}{3}x + 4 \) 
    D) \( y = \frac{3}{5}x - 4 \) 
     
  5. What is the slope of the line passing through (2,3) and (5,9)?
    A) 3 
    B) 2 
    C) 1 
    D) 4 
     
  6. Find the x-intercept of \( y = -2x + 6 \).
    A) 2 
    B) 3 
    C) -3 
    D) -2 
     
  7. Which equation is in slope-intercept form?
    A) \( 3x + 2y = 6 \) 
    B) \( y = -x + 4 \) 
    C) \( 4x - y = 7 \) 
    D) \( 2x - 3y = 9 \) 
     
  8. If a line has a slope of -2 and passes through (3,5), what is the y-intercept?
    A) 1 
    B) -1 
    C) 9 
    D) -9 
     
  9. Solve for x: \( 5x + 2 = 3x + 8 \).
    A) 2 
    B) 3 
    C) 5 
    D) 6 
     
  10. Which line is parallel to \( y = 2x + 3 \)?
    A) \( y = -2x + 1 \) 
    B) \( y = \frac{1}{2}x - 4 \) 
    C) \( y = 2x - 5 \) 
    D) \( y = -\frac{1}{2}x + 6 \) 
     

Answer Key (with Solutions)

  1. Solution for \( x \): \( 7x - 4 = 24 \)
    Add 4 to both sides: \( 7x = 28 \) 
    Divide by 7: \( x = 4 \) 
    Correct Answer: B
  2. Solution for \( x \): \( 3(x + 5) = 2x + 15 \)
    Distribute: \( 3x + 15 = 2x + 15 \) 
    Subtract \( 2x \) from both sides: \( x + 15 = 15 \) 
    Subtract 15: \( x = 0 \) 
    Correct Answer: B
  3. Solution: \( 4(2x - 3) + 6x \)
    Distribute: \( 8x - 12 + 6x \) 
    Combine like terms: \( 14x - 12 \) 
    Correct Answer: B
  4. Convert to slope-intercept form: \( 5x - 3y = 12 \)
    Subtract \( 5x \) from both sides: \( -3y = -5x + 12 \) 
    Divide by -3: \( y = \frac{5}{3}x - 4 \) 
    Correct Answer: A
  5. Find the slope of the line passing through (2,3) and (5,9)
    Use slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) 
    \( m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2 \) 
    Correct Answer: B
  6. Find the x-intercept of \( y = -2x + 6 \)
    Set \( y = 0 \) and solve for \( x \). 
    \( 0 = -2x + 6 \) → \( 2x = 6 \) → \( x = 3 \) 
    Correct Answer: B
  7. Which equation is in slope-intercept form?
    Correct form is \( y = mx + b \). 
    Only option B is written in this form. 
    Correct Answer: B
  8. If a line has a slope of -2 and passes through (3,5), what is the y-intercept?
    Use point-slope form: \( y - y_1 = m(x - x_1) \) 
    \( y - 5 = -2(x - 3) \) → \( y = -2x + 11 \) 
    Correct Answer: C