Integrated Math 1 Semester 2 Final Study Guide

Systems of Linear Equations: Solving by Graphing

• Solution Identification: The solution to a system of equations is the point of intersection $(x, y)$ where both lines meet on the coordinate plane.

• Problem 1: A system defined by the equations:

  • $y = -\frac{5}{3}x - 1$

  • $y = \frac{1}{3}x + 3$

• Problem 2: A system defined by the equations:

  • $y = x + 3$

  • $y = 5x - 1$

• Problem 3: A system defined by standard form equations:

  • $7x - 4y = -12$

  • $x - 4y = 12$

• Problem 4: A system involving a vertical line:

  • $x - y = -1$

  • $x = 3$

Systems of Linear Equations: Solving by Substitution

• Methodology: Solve one equation for a single variable and substitute that expression into the second equation.

• Problem 5:

  • Equation 1: $y = -4x + 6$

  • Equation 2: $-6x - 2y = -10$

• Problem 6:

  • Equation 1: $y = x + 2$

  • Equation 2: $4x - 3y = -7$

• Problem 7:

  • Equation 1: $-3x - 5y = -1$

  • Equation 2: $x + 3y = -5$

• Problem 8:

  • Equation 1: $4x + y = -8$

  • Equation 2: $-6x - 3y = 0$

Systems of Linear Equations: Solving by Elimination

• Methodology: Add or subtract equations to eliminate one variable, often requiring multiplication of one or both equations by a constant.

• Problem 9:

  • $-x + 4y = -26$

  • $10x - 4y = -28$

• Problem 10:

  • $x + 2y = 8$

  • $3x + 2y = -8$

• Problem 11:

  • $10x - 8y = -22$

  • $-20x + 6y = 4$

• Problem 12:

  • $12x - 10y = 4$

  • $-6x + 4y = -4$

Application of Linear Systems: Real-World Word Problems

• Problem 13: School Fundraiser (Wrapping Paper)

  • Scenario: John and Stefan are selling rolls of plain and holiday wrapping paper.

  • John's Sales: $5$ rolls of plain and $6$ rolls of holiday for a total of $\$136$.

  • Stefan's Sales: $7$ rolls of plain and $6$ rolls of holiday for a total of $\$152$.

  • Objective: Determine the cost of one roll of each type.

  • System Setup:

    • $5p + 6h = 136$

    • $7p + 6h = 152$

• Problem 14: Yard Improvement (Daylilies and Geraniums)

  • Scenario: Eugene and Jaidee bought supplies from the same store.

  • Eugene's Purchase: $6$ daylilies and $3$ geraniums for $\$69$.

  • Jaidee's Purchase: $3$ daylilies and $3$ geraniums for $\$48$.

  • Objective: Find the cost of one daylily and one geranium.

  • System Setup:

    • $6d + 3g = 69$

    • $3d + 3g = 48$

• Problem 15: School Fundraiser (Cheesecakes)

  • Scenario: Nicole and Aliyah are selling French silk and apple cheesecakes.

  • Nicole's Sales: $2$ French silk and $2$ apple cheesecakes for a total of $\$26$.

  • Aliyah's Sales: $6$ French silk and $1$ apple cheesecake for a total of $\$33$.

  • Objective: Determine the cost of one French silk and one apple cheesecake.

  • System Setup:

    • $2f + 2a = 26$

    • $6f + 1a = 33$

• Problem 16: Musical Ticket Sales

  • Scenario: Sales for senior citizen tickets and child tickets.

  • Day 1: $6$ senior tickets and $6$ child tickets for $\$90$.

  • Day 2: $5$ senior tickets and $2$ child tickets for $\$51$.

  • Objective: Determine the price of each ticket type.

Systems of Linear Inequalities: Graphical Solutions

• Methodology: Graph each inequality separately, using dashed lines for ($<$, $>$) and solid lines for ($\le$, $\ge$). The solution consists of the overlapping shaded region.

• Problem 17:

  • $y < x - 2$

  • $y > -\frac{4}{3}x + 3$

• Problem 18:

  • $x \le 2$

  • $y < -\frac{3}{2}x + 2$

• Problem 19:

  • $3x + 2y \ge 2$

  • $x - 2y > 6$

• Problem 20:

  • $3x + y \ge 3$

  • $x + 2y \le -4$

Geometric Sequences: Identification and Analysis

• Definition: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$).

• Sequence 21: $-4, 12, -36, 108, \dots$

  • Analysis: Multiplied by $-3$ consistently. Result: Geometric.

• Sequence 22: $2, -6, -14, -22, \dots$

  • Analysis: Decreases by sums, not factors. Result: Not Geometric.

• Problem 23: Sequence $-3, -12, -48, -192, \dots$

  • Common Ratio ($r$): $4$

  • Next Three Terms: $-768, -3072, -12288$

• Problem 24: Sequence $4, 8, 16, 32, \dots$

  • Common Ratio ($r$): $2$

  • Next Three Terms: $64, 128, 256$

Geometric Sequences: Formulas and Predictions

• Explicit Formula: $a_n = a_1 \times (r)^{n-1}$

• Recursive Formula: $a_n = a_{n-1} \times r$, with defined $a_1$

• Problem 25: $4, -24, 144, -864, \dots$

  • Common Ratio ($r$): $-6$

  • Explicit Formula: $a_n = 4 \times (-6)^{n-1}$

• Problem 26: $4, -20, 100, -500, \dots$

  • Common Ratio ($r$): $-5$

  • Explicit Formula: $a_n = 4 \times (-5)^{n-1}$

• Problem 27: $-3, 12, 48, 192, \dots$

  • Common Ratio ($r$): This sequence is inconsistent as written (first jump is $\times -4$, subsequent are $\times 4$).

• Problem 28: $2, 4, 8, -16, \dots$

  • Analysis: Search for 8th term based on explicit calculation.

• Problem 29: $4, 24, 144, 864, \dots$

  • Common Ratio ($r$): $6$

  • Recursive Formula: $a_n = a_{n-1} \times 6$

• Problem 30: $-3, -12, -48, -192, \dots$

  • Common Ratio ($r$): $4$

  • Recursive Formula: $a_n = a_{n-1} \times 4$

Exponential Functions: Graphing and Behavior

• Problem 31: $y = (\frac{1}{2})^x \times 5$

• Problem 32: $y = 2 \times 2^x$

• Problem 33: $y = -2 \times (\frac{1}{2})^x$

• Problem 34: $y = -3 \times 2^x - 2$

Geometric Transformations: Coordinate Rules and Descriptions

• Transformation Analysis (Items 53-56):

  • Descriptions include translations (shifts), reflections (flips), and rotations (turns).

• Reflections and Rotations:

  • Problem 47: Reflection across the line $y = -x$.

  • Problem 48: Reflection across the $y$-axis.

  • Problem 49: Reflection across the $x$-axis.

  • Problem 50: Reflection across the vertical line $x = 1$.

  • Problem 51: Rotation $180^{\circ}$ about the origin.

  • Problem 52: Rotation $90^{\circ}$ counterclockwise about the origin.

Congruence in Geometry: Triangle Correspondence

• Congruence Statements: Names the identical parts of two triangles.

• Problem 57: $\Delta DEF \cong \Delta DIJ$

  • Identify: $\angle F = ?$ (Answer: $\angle J$)

• Problem 58: $\Delta SRQ \cong \Delta RSV$

  • Identify: $RQ = ?$ (Answer: $SV$)

• Problem 59: $\Delta XWY \cong \Delta SQR$

  • Find length: $ZX = ?$

• Problem 60: $\Delta JKL \cong \Delta JKT$

  • Find length: $KL = ?$

Criteria for Triangle Congruence: Necessary Missing Information

• Requirement: State what additional information is required to prove congruence using a specific postulate.

• Problem 67: SAS (Side-Angle-Side)

  • Requires two sides and the included angle to be congruent.

• Problem 68: HL (Hypotenuse-Leg)

  • Applies only to right triangles; requires the hypotenuse and one leg to be congruent.

• Problem 69: ASA (Angle-Side-Angle)

  • Requires two angles and the side included between them to be congruent.

• Problem 70: SSS (Side-Side-Side)

  • Requires all three pairs of corresponding sides to be congruent.

Analysis of Triangle Congruence

• Process: Determine if two triangles are congruent based on given markings and state the reasoning (SSS, SAS, ASA, AAS, or HL).

• Markers (Items 61-66):

  • Tick marks on sides indicate congruent line segments.

  • Arcs on angles indicate congruent angles.

  • Right angle symbols indicate right triangles for HL possibility.

Linear Relationships: Parallel and Perpendicular Slopes

• Parallel Lines: Slopes ($m$) are identical ($m_1 = m_2$).

• Perpendicular Lines: Slopes are negative reciprocals ($m_1 \times m_2 = -1$).

• Problem 71: Find the slope of a line parallel to $y = \frac{3}{4}x + 3$

  • Parallel slope: $m = \frac{3}{4}$

• Problem 72: Find the slope of a line parallel to $4x - 3y = -9$

  • Rewrite as $y = \frac{4}{3}x + 3$. Parallel slope: $m = \frac{4}{3}$

• Problem 73: Find the slope of a line perpendicular to $y = 3x + 1$

  • Perpendicular slope: $m = -\frac{1}{3}$

• Problem 74: Find the slope of a line perpendicular to $3x - 2y = 4$

  • Rewrite as $y = \frac{3}{2}x - 2$. Perpendicular slope: $m = -\frac{2}{3}$