Integrated Math 1 Unit 1 Solving Linear Equations Study Notes

Integrated Math 1 Unit 1: Solving Linear Equations and Learning Targets

• Big Idea: Mathematical properties can be used to solve real-world problems.

• Integrated Math 1 Learning Objectives:

  • LT A: I can solve linear equations and justify reasoning using the distributive property and the properties of equality (addition, subtraction, multiplication, and division) (Unit 1.1).

  • LT B: I can use linear equations to solve real-life problems (Unit 1.1, 1.1a).

  • LT C: I can solve multi-step equations and use them to solve real-life problems (Unit 1.2, 1.2b, 1.2c).

  • LT D: I can solve special solutions of linear equations (Unit 1.3, 1.3a).

  • LT E: I can solve absolute value equations (Unit 1.4).

  • LT F: I can rewrite literal equations (Unit 1.5).

Mathematical Operations and Expressions Review

• Order of Operations (PEMDAS) Evaluation:

  • $7 - 2 + 5 = 5 + 5 = 10$

  • $(7 - 4)^2 + 6 \times 5 = (3)^2 + 30 = 9 + 30 = 39$

  • $\frac{8}{2} \times 4^2 = 4 \times 16 = 64$

• Integer Arithmetic:

  • $1 + (-3) = -2$

  • $0 + (-9) = -9$

  • $5 - (-3) = 5 + 3 = 8$

  • $-5 + (-8) = -13$

  • $15 \div (-3) = -5$

  • $-30 \div (-3) = 10$

  • $-2(13) = -26$

  • $-9(-5) = 45$

  • $\frac{-14}{-7} = 2$

• Adding and Subtracting Fractions:

  • $\frac{7}{8} - \frac{4}{5} = \frac{35}{40} - \frac{32}{40} = \frac{3}{40}$

  • $\frac{3}{4} + \frac{2}{3} = \frac{9}{12} + \frac{8}{12} = \frac{17}{12}$

  • $\frac{9}{10} - \frac{2}{5} = \frac{9}{10} - \frac{4}{10} = \frac{5}{10} = \frac{1}{2}$

  • $\frac{2}{7} + \frac{3}{2} = \frac{4}{14} + \frac{21}{14} = \frac{25}{14}$

Algebraic Foundations and Simplifying Expressions

• Combining Like Terms:

  • $2x + x = 3x$

  • $-x + x = 0$

  • $100 - 4x + 2x = 100 - 2x$

  • $3x - 2x + 4 - x = 3x - 3x + 4 = 4$

  • $-7 - 17 = -x$

  • $3x + 5x = 8x$

  • $4x + 2x - 6 = 6x - 6$

  • $3x + 4y + 2x - 2 = 5x + 4y - 2$

  • $15x - 3x + 4x - 12x = 19x - 15x = 4x$ (Wait, correct path: $3x + 4x - 2 \times 6x = 7x - 12x = -5x$)

• Distributive Property and Equation Simplification:

  • $2(9x - 2) + 4 = 18x - 4 + 4 = 18x$

Properties of Equality

• Addition Property of Equality (APE): Let $a$, $b$, and $c$ be real numbers. If $a = b$, then $a + c = b + c$.

• Subtraction Property of Equality (SPE): Let $a$, $b$, and $c$ be real numbers. If $a = b$, then $a - c = b - c$.

• Multiplication Property of Equality (MPE): Let $a$, $b$, and $c$ be real numbers. If $a = b$, then $a \times c = b \times c$.

• Division Property of Equality (DPE): Let $a$, $b$, and $c$ be real numbers. If $a \times c = b \times c$, then $a = b$.

Solving Linear Equations

• One-Step Equations:

  • $x + 4 = 16 \rightarrow x = 12$ (Justification through subtraction property of equality; check: $12 + 4 = 16, 16 = 16$).

  • $x + 7 = -12 \rightarrow x = -19$.

  • $-23 + x + 4 = 11 \rightarrow x - 19 = 11 \rightarrow x = 30$ (Wait, per transcript: $-23 + x + 4 = 4 \rightarrow -19 + x = 4 \rightarrow x = 23$). Corrected per page logic: $-23 + x + 4 = 4 \rightarrow x - 19 = 4 \rightarrow x = 23$. Actually, calculation shows $-23 + 27 = 4$, so check is $x = 27$.

  • $-21 = y - 11 \rightarrow y = -10$.

  • $5g = 20 \rightarrow g = 4$.

  • $-8y = 64 \rightarrow y = -8$.

  • $\frac{x}{-3} = 6 \rightarrow x = -18$.

  • $\frac{y}{7} = 10 \rightarrow y = 70$.

• Multi-Step Equations:

  • $2.5x - 13 = 2 \rightarrow 2.5x = 15 \rightarrow x = 6$.

  • $-12 = 9x - 6x + 15 \rightarrow -12 = 3x + 15 \rightarrow -27 = 3x \rightarrow x = -9$.

  • $-10x + 12 = 18 - 2x \rightarrow -8x = 6 \rightarrow x = -\frac{3}{4}$ (Simplified: $-12x + 12 = 18 \rightarrow -12x = 6 \rightarrow x = -\frac{1}{2}$).

  • $2(1 - x) + 3 = -8 \rightarrow 2 - 2x + 3 = -8 \rightarrow 5 - 2x = -8 \rightarrow -2x = -13 \rightarrow x = 6.5$.

  • $-4(2x + 5) - 3x = 35 \rightarrow -8x - 20 - 3x = 35 \rightarrow -11x = 55 \rightarrow x = -5$.

  • $15 = 5 + 4(2d - 3) \rightarrow 15 = 5 + 8d - 12 \rightarrow 15 = 8d - 7 \rightarrow 22 = 8d \rightarrow d = \frac{11}{4}$.

• Equations with Variables on Both Sides:

  • $10 - 4x = 9x \rightarrow 10 = 13x \rightarrow x = \frac{10}{13}$ (Wait, calculation on sheet: $10 - 4x = -9x \rightarrow 10 = -5x \rightarrow x = -2$).

  • $3(3x - 4) = \frac{1}{4}(32x + 56) \rightarrow 9x - 12 = 8x + 14 \rightarrow x = 26$.

  • $2x = 3x + 10 \rightarrow -x = 10 \rightarrow x = -10$.

  • $\frac{1}{2}(6h - 4) = -5h + 1 \rightarrow 3h - 2 = -5h + 1 \rightarrow 8h = 3 \rightarrow h = \frac{3}{8}$.

Identifying Solution Types

• One Solution: The end result is a single variable equal to a number, e.g., $x = 3$.

• No Solution: The variables cancel out completely, leaving an inequality that is false, e.g., $-1 = 3$ or $-14 = 18$.

• Many Solutions (Infinitely Many/All Solutions): The variables cancel out and leave an identity where both sides are equal, e.g., $4 = 4$ or $by - 6 = by - 6$.

• Examples of No/All Solutions:

  • $4(1 - p) = -4p + 4 \rightarrow 4 - 4p = -4p + 4 \rightarrow 4 = 4$ (All solutions).

  • $10x + 7 = 3 - 10x \rightarrow 20x = -4 \rightarrow x = -\frac{1}{5}$ (One solution).

  • $3(2y - 2) = 2(3y - 3) \rightarrow 6y - 6 = 6y - 6$ (All solutions).

  • $\frac{5}{6}m - 10 = \frac{5}{6} (m - 12) \rightarrow \frac{5}{6}m - 10 = \frac{5}{6}m - 10$ (All solutions).

Application of Geometry and Real-World Problems

• Quadrilateral Interior Angles: Sum of all interior angles of a quadrilateral is $360^{\circ}$.

  • Example A: Angles are $x, 80^{\circ}, 85^{\circ}, 100^{\circ}$. Equation: $80 + 85 + 100 + x = 360 \rightarrow x + 265 = 360 \rightarrow x = 95^{\circ}$.

  • Example B: Angles are $10x+7, 127, 5x+3, 88$. Equation: $(10x + 7) + 127 + (5x + 3) + 88 = 360 \rightarrow 15x + 225 = 360 \rightarrow 15x = 135 \rightarrow x = 9^{\circ}$.

• Banking: Checking account balance discrepency. Current balance is $\$68$. Bank states $\$26$. Equation: $68 = 26 + x$. The forgotten check was $\$42$.

• Flag Geometry: Length $(L)$ is $1.9$ times width $(W)$. If $L = 9.5\,ft$, equation is $9.5 = 1.9 \times W$. The width $W$ is $5\,ft$.

• Simple Interest: Formula is $I = prt$ (Interest = principle $\times$ rate $\times$ time). For principle $p = \$3600$, rate $r = 0.05$, and time $t = 9\,years$. $I = (3600)(0.05)(9) = \$1620$. Total balance including interest: $3600 + 1620 = \$5220$.

• Repair Costs: Total bill is $\$553$. Parts cost $\$265$ and labor is $\$48$ per hour. Equation: $553 = 265 + 48h$. Solving gives $288 = 48h$, resulting in $h = 6\,hours$ of labor.

• Meeting Distance: Two people are $190\,miles$ apart driving toward each other at $50\,mph$ and $45\,mph$. Equation: $50t + 45t = 190$. Solving gives $95t = 190$, so they meet in $t = 2\,hours$.

• Service Comparison: Company A charges $\$60.00$ installation + $\$42.95/month$. Company B charges $\$25.00$ installation + $\$49.95/month$. Equation: $60 + 42.95m = 25 + 49.95m \rightarrow 35 = 7m \rightarrow m = 5$. Costs are equal after $5$ months.

• Triangular Geometry: Sum of angles in a triangle is $180^{\circ}$. Angle measures given as $2k, k, 45^{\circ}$. Equation: $2k + k + 45 = 180 \rightarrow 3k = 135 \rightarrow k = 45^{\circ}$.

• Purchasing: Bouquet of $4$ roses and $8$ lilies for $\$40$. Roses cost $\$6$ each. Equation: $4(6) + 8l = 40 \rightarrow 24 + 8l = 40 \rightarrow 8l = 16 \rightarrow l = \$2$ per lily.

Absolute Value Equations

• Definition: Absolute value represents the distance from zero. It is always positive or zero ($|a| \geq 0$).

• Properties:

  1. $|a| \geq 0$ (Positive).

  2. $|-a| = |a|$ (Inverse values have equal absolute values).

  3. $|ab| = |a| \times |b|$

  4. $|\frac{a}{b}| = \frac{|a|}{|b|}$, provided $b \neq 0$.

• Solving Rules:

  • To solve $|ax + b| = c$, where $c \geq 0$, solve the two linear equations: $ax + b = c$ and $ax + b = -c$.

  • Isolation Principle: If there is a number outside the absolute value, it must be isolated first before splitting the equation. For example, in $|ax + b| + c = d$, isolate to $|ax + b| = d - c$ first.

  • Extraneous Solution: A solution that arrives from the solving process but does not satisfy the original absolute value equation.

• Examples:

  • $|x| = 5 \rightarrow x = 5, x = -5$.

  • $|x| = -2 \rightarrow$ Not possible (No solution).

  • $|x - 4| = 6 \rightarrow x - 4 = 6 \rightarrow x = 10$; $x - 4 = -6 \rightarrow x = -2$. Both are valid.

  • $3|x - 1| = -5 \rightarrow$ Absolute value cannot equal a negative after isolation, so no solution.

  • $|7x - 6| + 3 = 3 \rightarrow |7x - 6| = 0 \rightarrow 7x = 6 \rightarrow x = \frac{6}{7}$. One solution.

  • $|3x - 6| = 6 \rightarrow 3x - 6 = 6 \rightarrow 3x = 12 \rightarrow x = 4$; $3x - 6 = -6 \rightarrow 3x = 0 \rightarrow x = 0$.

  • $|3x + 9| - 10 = -4 \rightarrow |3x + 9| = 6 \rightarrow 3x + 9 = 6 \rightarrow x = -1$; $3x + 9 = -6 \rightarrow x = -5$.

  • $|2x + 12| = 4x$: Case 1: $2x + 12 = 4x \rightarrow 2x = 12 \rightarrow x = 6$. Case 2: $2x + 12 = -4x \rightarrow 6x = -12 \rightarrow x = -2$. Checking $x = -2 \rightarrow |2(-2) + 12| = 4(-2) \rightarrow |8| = -8$ (False). Solution is only $x = 6$.

Literal Equations

• Definition: An equation with two or more variables. Also commonly known as a formula.

• Rewriting Techniques: Solve for one specific variable in terms of the others.

• Examples:

  • $3y + 4x = 9$ for $y \rightarrow 3y = -4x + 9 \rightarrow y = -\frac{4}{3}x + 3$.

  • $y = 3x + 5x + z$ for $x \rightarrow y = 8x + z \rightarrow y - z = 8x \rightarrow x = \frac{y - z}{8}$.

  • $\frac{1}{2}x - 3 = m$ for $x \rightarrow \frac{1}{2}x = m + 3 \rightarrow x = 2m + 6$.

  • $3 + 5x - z - 3x = y$ for $x \rightarrow 2x + 3 - z = y \rightarrow 2x = y - 3 + z \rightarrow x = \frac{y - 3 + z}{2}$.

  • $9 - c = 17d$ for $c \rightarrow -c = 17d - 9 \rightarrow c = -17d + 9$.

  • $A = \frac{1}{2}h(b_1 + b_2)$ for $b_1 \rightarrow 2A = h(b_1 + b_2) \rightarrow \frac{2A}{h} = b_1 + b_2 \rightarrow b_1 = \frac{2A}{h} - b_2$.

  • $V = lhw$ for $w \rightarrow w = \frac{V}{lh}$.

  • $A = \frac{h(b + c)}{2}$ for $c \rightarrow 2A = h(b + c) \rightarrow \frac{2A}{h} = b + c \rightarrow c = \frac{2A}{h} - b$.

Practice Test Items

• Finding values in Formulas:

  • Given $D = \frac{1}{3}(P - 4)$, find $P$ when $D = 6$. Calculation: $3D = P - 4 \rightarrow 18 = P - 4 \rightarrow P = 22$.

  • Given $S = 2\pi r h$, find $h$ when $r = 2$ and $S = 48\pi$. Calculation: $h = \frac{S}{2\pi r} = \frac{48\pi}{2\pi(2)} = \frac{48}{4} = 12$.

• Jesse's Babysitting: Jesse earns $\$6/hr$. Last week earned $\$71$, which included a $\$20$ allowance. Equation: $6h + 20 = 71 \rightarrow 6h = 51 \rightarrow h = 8.5\,hours$.

• Slippers Sales: Some pairs sold at $\$6$, the rest at $\$11$. Total made is $\$1600$. $80$ pairs were the $\$11$ variety. Equation: $6A + 11(80) = 1600 \rightarrow 6A + 880 = 1600 \rightarrow 6A = 720 \rightarrow A = 120\,pairs$ at $\$6$.