Math Exam 2

Algebraic equations: solutions or classification

1a. Solve $4(x+2)-6(1-x)=624(x+2)\; –\; 6(1-x)\; =\; 62$

• Expand:

$$4x+8-6+6x=624x\; +\; 8\; –\; 6\; +\; 6x\; =\; 62$$

• Combine like terms:

$$10x+2=6210x\; +\; 2\; =\; 62$$

• Solve:

$$10x=60\Rightarrow x=610x\; =\; 60\; \backslash Rightarrow\; x\; =\; 6$$

1b. Solve $5x+3-2x=-4+10x+7-7x5x\; +\; 3\; –\; 2x\; =\; -4\; +\; 10x\; +\; 7\; –\; 7x$

• Simplify both sides:

$$\text{LHS: }3x+3,\text{RHS: }3x+3\backslash text\{LHS:\; \}\; 3x\; +\; 3,\backslash quad\; \backslash text\{RHS:\; \}\; 3x\; +\; 3$$

• Conclusion: Identity; true for all real $xx$.

1c. Solve $7(x-2)-3x=4x7(x-2)\; –\; 3x\; =\; 4x$

• Expand and simplify:

$$7x-14-3x=4x\Rightarrow 4x-14=4x7x\; –\; 14\; –\; 3x\; =\; 4x\; \backslash Rightarrow\; 4x\; –\; 14\; =\; 4x$$

• Subtract $4x4x$ from both sides:

$$-14=0-14\; =\; 0$$

• Conclusion: No solution (contradiction).

1d. Solve $8x+t-4=6(x-t)8x\; +\; t\; –\; 4\; =\; 6(x\; –\; t)$

• Expand RHS:

$$8x+t-4=6x-6t8x\; +\; t\; –\; 4\; =\; 6x\; –\; 6t$$

• Collect terms in $xx$:

$$2x+7t-4=0\Rightarrow 2x=4-7t2x\; +\; 7t\; –\; 4\; =\; 0\; \backslash Rightarrow\; 2x\; =\; 4\; –\; 7t$$

• Solve:

$$x=\frac{4-7t}{2}x\; =\; \backslash frac\{4\; –\; 7t\}\{2\}$$

Rates: scoring averages

2a. Points per minute

• Compute:

$$\frac{32}{40}=0.8\text{ points per minute}\backslash frac\{32\}\{40\}\; =\; 0.8\; \backslash text\{\; points\; per\; minute\}$$

2b. Minutes per point

• Compute:

$$\frac{40}{32}=1.25\text{ minutes per point}\backslash frac\{40\}\{32\}\; =\; 1.25\; \backslash text\{\; minutes\; per\; point\}$$

Linear equation forms

3. Three of the five common forms

• Slope–intercept form: $y=mx+by\; =\; mx\; +\; b$
• Point–slope form: $y-y_1=m(x-x_1)y\; –\; y\_1\; =\; m(x\; –\; x\_1)$
• Standard form: $Ax+By=C$
• Karen’s altitude: rate of change analysis
• 4a. Interval of fastest increase
• Compute slopes over 15-minute intervals:
  • 0–15: $\frac{680-540}{15}=9.\bar{3}\backslash frac\{680\; –\; 540\}\{15\}\; =\; 9.\backslash overline\{3\}$
  • 15–30: $\frac{1640-680}{15}=64\backslash frac\{1640\; –\; 680\}\{15\}\; =\; 64$
  • 30–45: $\frac{2000-1640}{15}=24\backslash frac\{2000\; –\; 1640\}\{15\}\; =\; 24$
  • 60–75: $00$
  • 75–90: $\frac{2200-1800}{15}\approx 26.67\backslash frac\{2200\; –\; 1800\}\{15\}\; \backslash approx\; 26.67$
• Conclusion: Fastest increase from 15 to 30 minutes.
• 4b. Average rate of change during fastest increase
• Compute:
• $$\frac{1640-680}{15}=64\text{ ft/min}\backslash frac\{1640\; –\; 680\}\{15\}\; =\; 64\; \backslash text\{\; ft/min\}$$
• 4c. Interval of fastest decrease
• Check decreasing intervals:
  • 45–60: $\frac{1800-2000}{15}=-13.\bar{3}\backslash frac\{1800\; –\; 2000\}\{15\}\; =\; -13.\backslash overline\{3\}$
  • 90–105: $\frac{1600-2200}{15}=-40\backslash frac\{1600\; –\; 2200\}\{15\}\; =\; -40$
  • 105–120: $\frac{420-1600}{15}\approx -78.67\backslash frac\{420\; –\; 1600\}\{15\}\; \backslash approx\; -78.67$
• Conclusion: Fastest decrease from 105 to 120 minutes.
• Lines: slopes, intercepts, and factoring
• 5a. Slope and intercept for $y-5=3(x+2)y\; –\; 5\; =\; 3(x\; +\; 2)$
• Rewrite:
• $$y=3x+6+5=3x+11y\; =\; 3x\; +\; 6\; +\; 5\; =\; 3x\; +\; 11$$
• Slope: $m=3m\; =\; 3$
• Y-intercept: $b=11b\; =\; 11$
• 5b. Slope and intercept for $y=-x+7y\; =\; -x\; +\; 7$
• Slope: $m=-1m\; =\; -1$
• Y-intercept: $b=7b\; =\; 7$
• 6a. Factor $3y+9x3y\; +\; 9x$
• Common factor: 3(y+3x)
• 6b. Factor $14ab+28ac14ab\; +\; 28ac$
• Common factor: 14a(b+2c)
• Writing linear equations from intercepts or slope
• 7a. Line with $xx$-intercept $-5-5$ and $yy$-intercept $-2-2$
• Points: $(-5,0)(-5,\; 0)$ and $(0,-2)(0,\; -2)$
• Slope:
• $$m=\frac{-2-0}{0-(-5)}=-\frac{2}{5}m\; =\; \backslash frac\{-2\; –\; 0\}\{0\; –\; (-5)\}\; =\; -\backslash frac\{2\}\{5\}$$
• Equation: y=−25x−2
• 7b. Line with slope $\frac{1}{2}\backslash frac\{1\}\{2\}$ and $yy$-intercept $44$
• Equation: y=12x+4
• Polynomial names, products, and factoring
• 8. Name by degree and number of terms
• a) $x^3+x+3x^3\; +\; x\; +\; 3$: Cubic trinomial
• b) $y-1y\; –\; 1$: Linear binomial
• c) $d^{5}d^5$: Quintic monomial
• d) $4x^2-3x+124x^2\; –\; 3x\; +\; 12$: Quadratic trinomial
• 9. Product $(x+3)(x-1)(x-5)(x\; +\; 3)(x\; –\; 1)(x\; –\; 5)$
• First multiply: (x+3)(x−1)=x2+2x−3
• Then: (x2+2x−3)(x−5)=x3−3×2−13x+15
• 10a. Factor $x^2-8x+16x^2\; –\; 8x\; +\; 16$
• Perfect square: (x−4)2
• 10c. Factor $4x^2-364x^2\; –\; 36$
• Difference of squares: 4(x2−9)=4(x−3)(x+3)
• Completing the square and solving
• 11. Solve $x^2-4x+4=12x^2\; –\; 4x\; +\; 4\; =\; 12$
• Recognize square:
• $$(x-2{)}^2=12(x\; –\; 2)^2\; =\; 12$$
• Take square roots:
• $$x-2=\pm \sqrt{12}=\pm 2\sqrt{3}x\; –\; 2\; =\; \backslash pm\; \backslash sqrt\{12\}\; =\; \backslash pm\; 2\backslash sqrt\{3\}$$
• Solutions:
• $$x=2\pm 2\sqrt{3}x\; =\; 2\; \backslash pm\; 2\backslash sqrt\{3\}$$
• Solution set: $\{2-2\sqrt{3},2+2\sqrt{3}\}\backslash \{\backslash ,2\; –\; 2\backslash sqrt\{3\},\backslash ;\; 2\; +\; 2\backslash sqrt\{3\}\backslash ,\backslash \}$
• Rational numbers
• 12. Definition
• Rational number: A number expressible as $\frac{p}{q}\backslash frac\{p\}\{q\}$ with integers $p,qp,\; q$ and $q\ne 0q\; \backslash neq\; 0$; its decimal expansion terminates or repeats.
• Work rate word problem
• 13a. Expressions for number solved over time
• Let $tt$ be time in hours since starting:
  • Katy: $K(t)=3+26tK(t)\; =\; 3\; +\; 26t$
  • Timmy: $T(t)=8+20tT(t)\; =\; 8\; +\; 20t$
• 13b. Number solved in 30 minutes
• Use $t=0.5t\; =\; 0.5$:
• $$K(0.5)=3+26(0.5)=16,T(0.5)=8+20(0.5)=18K(0.5)\; =\; 3\; +\; 26(0.5)\; =\; 16,\backslash quad\; T(0.5)\; =\; 8\; +\; 20(0.5)\; =\; 18$$
• 13c. Time until equal and how many equations that is
• Set equal:
• $$3+26t=8+20t\Rightarrow 6t=5\Rightarrow t=\frac{5}{6}\text{ hours}=50\text{ minutes}3\; +\; 26t\; =\; 8\; +\; 20t\; \backslash Rightarrow\; 6t\; =\; 5\; \backslash Rightarrow\; t\; =\; \backslash frac\{5\}\{6\}\; \backslash text\{\; hours\}\; =\; 50\; \backslash text\{\; minutes\}$$
• Number solved then:
• $$K\text{\hspace{0.17em}⁣}\left(\frac{5}{6}\right)=3+26\text{\hspace{0.17em}⁣}\left(\frac{5}{6}\right)=\frac{74}{3}\approx 24.67K\backslash !\backslash left(\backslash frac\{5\}\{6\}\backslash right)\; =\; 3\; +\; 26\backslash !\backslash left(\backslash frac\{5\}\{6\}\backslash right)\; =\; \backslash frac\{74\}\{3\}\; \backslash approx\; 24.67$$
• 13d. Who finishes first if each must solve 27 total
• Katy needs $27-3=2427\; –\; 3\; =\; 24$ more: time $=\frac{24}{26}=\frac{12}{13}\approx 0.923=\; \backslash frac\{24\}\{26\}\; =\; \backslash frac\{12\}\{13\}\; \backslash approx\; 0.923$ hours $\approx 55.38\backslash approx\; 55.38$ minutes.
• Timmy needs $27-8=1927\; –\; 8\; =\; 19$ more: time $=\frac{19}{20}=0.95=\; \backslash frac\{19\}\{20\}\; =\; 0.95$ hours $\approx 57\backslash approx\; 57$ minutes.
• Conclusion: Katy finishes first (by about 1.6 minutes).$$$$