5.10: Exercise Supplement
Exercise Supplement
Solving Equations - Further Techniques in Equation Solving
Solve the equations for the following problems.
\(y+3=11\)
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\(y=8\)
\(a−7=4\)
\(r−1=16\)
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\(r=17\)
\(a+2=0\)
\(x+6=−4\)
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\(x=−10\)
\(x−5=−6\)
\(x+8=8\)
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\(x=0\)
\(y−4=4\)
\(2x=32\)
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\(x=16\)
\(4x=24\)
\(3r=−12\)
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\(r=−4\)
\(6m=−30\)
\(−5x=−30\)
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\(x=6\)
\(−8y=−72\)
\(−x=6\)
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\(x=−6\)
\(−y=−10\)
\(3x+7=19\)
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\(x=4\)
\(6x−1=29\)
\(4x+2=−2\)
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\(x=−1\)
\(6x−5=−29\)
\(8x+6=−10\)
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\(x=−2\)
\(9a+5=−22\)
\(\dfrac{m}{6} + 4 = 8\)
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\(m=24\)
\(\dfrac{b}{5} - 2 = 5\)
\(\dfrac{y}{9} = 54\)
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\(y=486\)
\(\dfrac{a}{-3} = -17\)
\(\dfrac{3a}{4} = 9\)
\(\dfrac{4y}{5} = -12\)
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\(y=−15\)
\(\dfrac{r}{4} = 7\)
\(\dfrac{6a}{-5} = 11\)
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\(a = -\dfrac{55}{6}\)
\(\dfrac{9x}{7} = 6\)
\(\dfrac{c}{2} - 8 = 0\)
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\(c=16\)
\(\dfrac{m}{-5} + 4 = -1\)
\(\dfrac{x}{7} - 15 = -11\)
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\(x=28\)
\(\dfrac{3x}{4} + 2 = 14\)
\(\dfrac{3r+2}{5} = -1\)
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\(r = -\dfrac{7}{3}\)
\(\dfrac{6x-1}{7} = -3\)
\(\dfrac{4x-3}{6} + 2 = -6\)
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\(x = -\dfrac{45}{4}\)
\(\dfrac{y-21}{8} = -3\)
\(4(x+2)=20\)
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\(x=3\)
\(−2(a−3)=16\)
\(−7(2a−1)=63\)
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\(a=−4\)
\(3x+7=5x−21\)
\(−(8r+1)=33\)
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\(r = -\dfrac{17}{4}\)
Solve \(I=prt\) for \(t\). Find the value of \(t\) when \(I=3500, P=3000\), and \(r=0.05\).
Solve \(A=LW\) for \(W\). Find the value of \(W\) when \(A=26\) and \(L=2\).
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\(W=13\)
Solve \(p=mv\) for \(m\). Find the value of m when \(p=4240\) and \(v=260\).
Solve \(P=R−C\) for \(R\). Find the value of \(R\) when \(P=480\) and \(C=210\).
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\(R=690\)
Solve \(P = \dfrac{nRT}{V}\) for \(n\).
Solve \(y=5x+8\) for \(x\).
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\(x = \dfrac{y-8}{5}\)
Solve \(3y−6x=12\) for \(y\).
Solve \(4y+2x+8=0\) for \(y\).
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\(y = -\dfrac{1}{2}x - 2\)
Solve \(k = \dfrac{4m + 6}{7}\) for \(m\)
Solve \(t = \dfrac{10a-3b}{2c}\) for \(b\)
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\(b = -\dfrac{2ct - 10a}{3}\)
Application I - Translating from Verbal to Mathematical Expressions
For the following problems, translate the phrases or sentences to mathematical expressions or equations.
A quantity less eight.
A number, times four plus seven.
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\(x(4+7)\)
Negative ten minus some number.
Two fifths of a number minus five.
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\(\dfrac{2}{5}x - 5\)
One seventh of a number plus two ninths of the number.
Three times a number is forty.
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\(3x=40\)
Twice a quantity plus nine is equal to the quantity plus sixty.
Four times a number minus five is divided by seven. The result is ten more than the number.
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\(\dfrac{(4x-5)}{7} = x + 10\)
A number is added to itself five times, and that result is multiplied by eight. The entire result is twelve.
A number multiplied by eleven more than itself is six.
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\(x(x+11)=6\)
A quantity less three is divided by two more than the quantity itself. The result is one less than the original quantity.
A number is divided by twice the number, and eight times the number is added to that result. The result is negative one.
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\(\dfrac{x}{2x} + 8x = -1\)
An unknown quantity is decreased by six. This result is then divided by twenty. Ten is subtracted from this result and negative two is obtained.
One less than some number is divided by five times the number. The result is the cube of the number.
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\(\dfrac{x-1}{5x} = x^3\)
Nine less than some number is multiplied by the number less nine. The result is the square of six times the number.
Application II - Solving Problems
For the following problems, find the solution.
This year an item costs $106, an increase of $10 over last year’s price. What was last year’s price?
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last year's price=$96
The perimeter of a square is 44 inches. Find the length of a side.
Nine percent of a number is 77.4. What is the number?
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\(x=860\)
Two consecutive integers sum to 63. What are they?
Four consecutive odd integers add to 56. What are they?
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\(x=11\)
\(x+2=13\)
\(x+4=15\)
\(x+6=17\)
If twenty-one is subtracted from some number and that result is multiplied by two, the result is thirty-eight. What is the number?
If 37% more of a quantity is 159.1, what is the quantit?
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\(x=116.13139\)
A statistician is collecting data to help her estimate the number of pickpockets in a certain city. She needs \(108\) pieces of data and is \(\dfrac{3}{4}\) done. How many pieces of data has she collected?
The statistician in problem 78 is eight pieces of data short of being \(\dfrac{5}{6}\) done. How many pieces of data has she collected?
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82 pieces of data
A television commercial advertises that a certain type of light bulb will last, on the average, 200 hours longer than three times the life of another type of bulb. If consumer tests show that the advertised bulb lasts 4700 hours, how many hours must the other type of bulb last for the advertiser’s claim to be valid?
Linear inequalities in One Variable
Solve the inequalities for the following problems.
\(y+3<15\)
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\(y<12\)
\(x−6≥12\)
\(4x+3>23\)
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\(x>5\)
\(5x−14<1\)
\(6a−6≤−27\)
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\(a \le -\dfrac{7}{2}\)
\(−2y≥14\)
\(−8a≤−88\)
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\(a≥11\)
\(\dfrac{x}{7} > -2\)
\(\dfrac{b}{-3} \le 4\)
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\(b≥−12\)
\(\dfrac{2a}{7} < 6\)
\(\dfrac{16c}{3} \ge -48\)
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\(c≥−9\)
\(−4c+3≤5\)
\(−11y+4>15\)
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\(y<−1\)
\(3(4x−5)>−6\)
\(−7(8x+10)+2<−32\)
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\(x > -\dfrac{9}{14}\)
\(5x+4≥7x+16\)
\(−x−5<3x−11\)
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\(x > \dfrac{3}{2}\)
\(4(6x+1)+2≥−3(x−1)+4\)
\(−(5x+6)+2x−1<3(1−4x)+11\)
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\(x < \dfrac{7}{3}\)
What numbers satisfy the condition: nine less than negative four times a number is strictly greater than negative one?
Linear Equations in Two Variables
Solve the equations for the following problems.
\(y=−5x+4\), if \(x=−3\)
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\((−3,19)\)
\(y=−10x+11\), if \(x=−1\)
\(3a+2b=14\), if \(b=4\)
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\((2,4)\)
\(4m+2k=30\), if \(m=8\)
\(−4r+5s=−16\), if \(s=0\)
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\((4,0)\)
\(y=−2(7x−4), if x=−1\)
\(−4a+19=2(b+6)−5\), if \(b=−1\)
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\((\dfrac{7}{2},−1)\)
\(6(t+8)=−(a−5)\), if \(a=10\)
\(−(a+b)=5\), if \(a=−5\)
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\((−5,0)\)
\(−a(a+1)=2b+1\), if \(a=−2\)