Simultaneous Equations of the First Degree

Excerpts

Simultaneous Equations of the First Degree
Independent equations involving the *same* unknown numbers are called **simultaneous equations**.
Simultaneous Equations of the First Degree
If we have two unknown numbers and but one relation between them, we can find an unlimited number of pairs of values for which the given relation will hold true.
Simultaneous Equations of the First Degree
If we have two unknown numbers, and two independent equations involving them, there is but *one* pair of values which will hold true for both equations.
Simultaneous Equations of the First Degree
Multiply the equations by such numbers as will make the coefficients of one of the unknown numbers equal in the resulting equations.
Simultaneous Equations of the First Degree
It is generally best to select the letter to be eliminated which requires the smallest multipliers to make its coefficients equal; and the smallest multiplier for each equation is found by dividing the L. C. M. of the coefficients of this letter by the given coefficient in that equation.
Simultaneous Equations of the First Degree
The expression $64$ means $60 + 4$, that is, $10$ *times* $6 + 4$, and has for its *digits* $6$ and $4$.
Simultaneous Equations of the First Degree
The interest for one year is $\dfrac{y}{100}$ of the principal; that is, $\dfrac{xy}{100}$.
Simultaneous Equations of the First Degree
To find $x$ in problem 26, add the equations; to find $y$, subtract one from the other. Do not clear of fractions.

Equations

Simultaneous Equations of the First Degree
x + y = 10
One relation between two unknown numbers leaves infinitely many pairs of values that satisfy it.
Simultaneous Equations of the First Degree
y = 10 - x
Solving x + y = 10 for y gives y as 10 minus x, so any chosen x determines y.
Simultaneous Equations of the First Degree
x - y = 2
A second equation expressing a relation between x and y different from x + y = 10, so the two together are independent equations.
Simultaneous Equations of the First Degree
x + y &= 10
The sum of the two digits of the number is 10.
Simultaneous Equations of the First Degree
10x + y + 18 &= 10y + x
Adding 18 to the number 10x + y gives the number with its digits reversed, 10y + x.
Simultaneous Equations of the First Degree
x - y &= -2
Dividing the reduced equation 9x - 9y = -18 by 9 gives x - y = -2.
Simultaneous Equations of the First Degree
y + 10 &= 3(x - 10)
After A gives B $10, B's money equals three times A's money.
Simultaneous Equations of the First Degree
x + 10 &= 2(y - 10)
After B gives A $10, A's money equals twice B's money.

Problems

Exercise 65

Exercise 65, problem 1, p. 124
-0.5$\left.\begin{alignedat}{2} 5x &+ 4y &&= 14 \\ 17x &- 3y &&= 31 \end{alignedat}\right\}$
Printed answer:
- $x = 2$, $y = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 2, y: 1}
Exercise 65, problem 10, p. 124
-0.5$\left.\begin{alignedat}{2} 6x &- 13y &&= -1 \\ 5x &- 12y &&= -2 \end{alignedat}\right\}$
Printed answer:
- $x = 2$, $y = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 2, y: 1}
Exercise 65, problem 11, p. 124
-0.5$\left.\begin{alignedat}{2} 7x &+ \Z y &&= 265 \\ 3x &- 5y &&= \Z\Z5 \end{alignedat}\right\}$
Printed answer:
- $x = 35$, $y = 20$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 35, y: 20}
Exercise 65, problem 12, p. 124
-0.5$\left.\begin{alignedat}{2} 2x &+ 3y &&= \Z7 \\ 8x &- 5y &&= 11 \end{alignedat}\right\}$
Printed answer:
- $x = 2$, $y = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 2, y: 1}
Exercise 65, problem 13, p. 124
-0.5$\left.\begin{alignedat}{2} 5x &+ 7y &&= 19 \\ 7x &+ 4y &&= 15 \end{alignedat}\right\}$
Printed answer:
- $x = 1$, $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 1, y: 2}
Exercise 65, problem 14, p. 124
-0.5$\left.\begin{alignedat}{2} 11x &- 12y &&= \Z9 \\ 4x &+ \Z5y &&= 22 \end{alignedat}\right\}$
Printed answer:
- $x = 3$, $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 3, y: 2}
Exercise 65, problem 15, p. 124
-0.5$\left.\begin{alignedat}{2} x &+ 8y &&= 17 \\ 7x &- 3y &&= \Z1 \end{alignedat}\right\}$
Printed answer:
- $x = 1$, $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 1, y: 2}
Exercise 65, problem 16, p. 124
-0.5$\left.\begin{alignedat}{2} 4x &+ 3y &&= 25 \\ 5x &- 4y &&= \Z8 \end{alignedat}\right\}$
Printed answer:
- $x = 4$, $y = 3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 4, y: 3}
Exercise 65, problem 17, p. 124
-0.5$\left.\begin{alignedat}{2} \frac{2x}{3} &- \frac{5y}{4} &&= 3 \\ \frac{7x}{4} &- \frac{5y}{3} &&= \frac{43}{3} \end{alignedat}\right\}$
Printed answer:
- $x = 12$, $y = 4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 12, y: 4}
Exercise 65, problem 18, p. 124
-0.5$\left.\begin{alignedat}{2} \frac{7x}{6} &+ \frac{6y}{7} &&= 32 \\ \frac{5x}{4} &- \frac{2y}{3} &&= 1 \end{alignedat}\right\}$
Printed answer:
- $x = 12$, $y = 21$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 12, y: 21}
Exercise 65, problem 19, p. 124
-0.5$\left.\begin{aligned} \frac{x + y}{4} - \frac{7x - 5y}{11} &= 3 \\ \frac{x}{5} - \frac{2y}{7} + 1 &= 0 \end{aligned}\right\}$
Printed answer:
- $x = 5$, $y = 7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 5, y: 7}
Exercise 65, problem 2, p. 124
-0.5$\left.\begin{alignedat}{2} 3x &- 2y &&= \Z5 \\ 2x &+ 5y &&= 16 \end{alignedat}\right\}$
Printed answer:
- $x = 3$, $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 3, y: 2}
Exercise 65, problem 20, p. 124
-0.5$\left.\begin{aligned} \frac{ 6x + 7y}{2} &= 22 \\ \frac{55y - 2x}{5} &= 20 \end{aligned}\right\}$
Printed answer:
- $x = 5$, $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 5, y: 2}
Exercise 65, problem 21, p. 124
-0.5$\left.\begin{alignedat}{2} \frac{x + y}{2} &- \frac{x - y}{3} &&= \Z8 \\ \frac{x + y}{3} &+ \frac{x - y}{4} &&= 11 \end{alignedat}\right\}$
Printed answer:
- $x = 18$, $y = 6$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 18, y: 6}
Exercise 65, problem 22, p. 124
-0.5$\left.\begin{aligned} \frac{8x - 5y}{7} + \frac{11y - 4x}{5} &= 4 \\ \frac{17x - 13y}{5} + \frac{2x}{3} &= 7 \end{aligned}\right\}$
Printed answer:
- $x = 3$, $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 3, y: 2}
Exercise 65, problem 23, p. 124
-0.5$\left.\begin{alignedat}{2} \frac{ 5x - 3y}{3} &+ \frac{7x - 5y}{11} &&= 4 \\ \frac{15y - 3x}{7} &+ \frac{7y - 3x}{5} &&= 4 \end{alignedat}\right\}$
Printed answer:
- $x = 3$, $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 3, y: 2}
Exercise 65, problem 24, p. 124
-0.5$\left.\begin{aligned} \frac{2x - 3}{4} - \frac{y - 8}{5} &= \frac{y + 3}{4} \\ \frac{x - 7}{3} + \frac{4y + 1}{11} &= 3 \end{aligned}\right\}$
Printed answer:
- $x = 7$, $y = 8$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 7, y: 8}
Exercise 65, problem 25, p. 124
-0.5$\left.\begin{alignedat}{2} \frac{x - 2y}{6} &- \frac{x + 3y}{4} &&= \frac{3}{2} \\ \frac{2x - y}{6} &- \frac{3x + y}{4} &&= \frac{5y}{4} \end{alignedat}\right\}$
Printed answer:
- $x = 8$, $y = -2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 8, y: -2}
Exercise 65, problem 26, p. 124
-0.5$\left.\begin{alignedat}{2} \frac{x}{a + b} &+ \frac{y}{a - b} &&= \frac{1}{a - b} \\ \frac{x}{a + b} &- \frac{y}{a - b} &&= \frac{1}{a + b} \end{alignedat}\right\}$
Printed answer:
- $x = \dfrac{a}{(a - b)}$, $y = \dfrac{b}{(a + b)}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: a/(a - b), y: b/(a + b)}
Exercise 65, problem 3, p. 124
-0.5$\left.\begin{alignedat}{2} 2x &- 3y &&= \Z7 \\ 5x &+ 2y &&= 27 \end{alignedat}\right\}$
Printed answer:
- $x = 5$, $y = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 5, y: 1}
Exercise 65, problem 4, p. 124
-0.5$\left.\begin{alignedat}{2} 7x &+ 6y &&= 20 \\ 2x &+ 5y &&= \Z9 \end{alignedat}\right\}$
Printed answer:
- $x = 2$, $y = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 2, y: 1}
Exercise 65, problem 5, p. 124
-0.5$\left.\begin{alignedat}{2} x &+ 5y &&= 11 \\ 3x &+ 2y &&= \Z7 \end{alignedat}\right\}$
Printed answer:
- $x = 1$, $y = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 1, y: 2}
Exercise 65, problem 6, p. 124
-0.5$\left.\begin{alignedat}{2} 3x &- 5y &&= 13 \\ 4x &- 7y &&= 17 \end{alignedat}\right\}$
Printed answer:
- $x = 6$, $y = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 6, y: 1}
Exercise 65, problem 7, p. 124
-0.5$\left.\begin{alignedat}{2} 8x &- \Z y &&= \Z3 \\ 7x &+ 2y &&= 63 \end{alignedat}\right\}$
Printed answer:
- $x = 3$, $y = 21$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 3, y: 21}
Exercise 65, problem 8, p. 124
-0.5$\left.\begin{alignedat}{2} 5x &- 4y &&= \Z7 \\ 7x &+ 3y &&= 70 \end{alignedat}\right\}$
Printed answer:
- $x = 7$, $y = 7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 7, y: 7}
Exercise 65, problem 9, p. 124
-0.5$\left.\begin{alignedat}{2} x &+ 21y &&= \Z2 \\ 2x &+ 27y &&= 19 \end{alignedat}\right\}$
Printed answer:
- $x = 23$, $y = -1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 23, y: -1}

Exercise 66

Exercise 66, problem 1, p. 127
If A gives B $$200$, A will then have half as much money as B; but if B gives A $$200$, B will have one-third as much as A. How much has each?
Printed answer:
- A, $$520$; B, $$440$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {a: 520, b: 440}
Exercise 66, problem 2, p. 127
Half the sum of two numbers is $20$, and three times their difference is $18$. Find the numbers.
Printed answer:
- $23$ and $17$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 23, y: 17}
Exercise 66, problem 3, p. 127
The sum of two numbers is $36$, and their difference is equal to one-eighth of the smaller number increased by $2$. Find the numbers.
Printed answer:
- $20$ and $16$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 20, y: 16}
Exercise 66, problem 4, p. 127
If $4$ yards of velvet and $3$ yards of silk are sold for $$33$, and $5$ yards of velvet and $6$ yards of silk for $$48$, what is the price per yard of the velvet and of the silk?
Printed answer:
- Velvet, $$6$; silk, $$3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {v: 6, s: 3}
Exercise 66, problem 5, p. 127
If $7$ bushels of wheat and $10$ of rye are sold for $$15$, and $4$ bushels of wheat and $5$ of rye are sold for $$8$, what is the price per bushel of the wheat and of the rye?
Printed answer:
- Wheat, $$1$; rye, $$\frac{4}{5}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {w: 1, r: Rational(4, 5)}
Exercise 66, problem 6, p. 127
If $12$ pounds of tea and $4$ pounds of coffee cost $$7$, and $4$ pounds of tea and $12$ pounds of coffee cost $$5$, what is the price per pound of tea and of coffee?
Printed answer:
- Tea, $$\frac{1}{2}$; coffee, $$\frac{1}{4}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {t: Rational(1, 2), c: Rational(1, 4)}
Exercise 66, problem 7, p. 127
Six horses and $7$ cows can be bought for $$1000$, and $11$ horses and $13$ cows for $$1844$. Find the value of a horse and of a cow.
Printed answer:
- Horse, $$92$; cow, $$64$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {h: 92, c: 64}

Exercise 67

Exercise 67, problem 1, p. 128
If the numerator of a certain fraction is increased by $2$ and its denominator diminished by $2$, its value will be $1$. If the numerator is increased by the denominator and the denominator is diminished by $5$, its value will be $5$. Find the fraction.
Printed answer:
- $\frac{3}{7}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 3, y: 7}
Exercise 67, problem 2, p. 128
If $1$ is added to the denominator of a fraction, its value will be $\frac{1}{2}$. If $2$ is added to its numerator, its value will be $\frac{3}{5}$. Find the fraction.
Printed answer:
- $\frac{13}{25}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 13, y: 25}
Exercise 67, problem 3, p. 128
If $1$ is added to the numerator of a fraction, its value will be $\frac{1}{5}$. If $1$ is added to its denominator, its value will be $\frac{1}{7}$. Find the fraction.
Printed answer:
- $\frac{3}{20}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 3, y: 20}
Exercise 67, problem 4, p. 128
If the numerator of a fraction is doubled and its denominator diminished by $1$, its value will be $\frac{1}{2}$. If its denominator is doubled and its numerator increased by $1$, its value will be $\frac{1}{7}$. Find the fraction.
Printed answer:
- $\frac{5}{21}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 5, y: 21}
Exercise 67, problem 5, p. 128
In a certain proper fraction the difference between the numerator and the denominator is $15$. If the numerator is multiplied by $4$ and the denominator increased by $6$, its value will be $1$. Find the fraction.
Printed answer:
- $\frac{7}{22}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 7, y: 22}
Exercise 67, problem None, p. 128
A certain fraction becomes equal to $\frac{1}{2}$ if $2$ is added to its numerator, and equal to $\frac{1}{3}$ if $3$ is added to its denominator. Find the fraction.
Printed answer:
- (none printed)
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 7, y: 18}

Exercise 68

Exercise 68, problem 1, p. 129
The sum of the two digits of a number is $9$, and if $9$ is added to the number, the digits will be reversed. Find the number.
Printed answer:
- $45$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {N: 45, x: 4, y: 5}
Exercise 68, problem 2, p. 129
A certain number of two digits is equal to eight times the sum of its digits, and if $45$ is subtracted from the number, the digits will be reversed. Find the number.
Printed answer:
- $72$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {N: 72, x: 7, y: 2}
Exercise 68, problem 3, p. 129
The sum of a certain number of two digits and the number formed by reversing the digits is $132$, and the difference of these numbers is $18$. Find the numbers.
Printed answer:
- $75$ and $57$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {N: 75, M: 57, x: 7, y: 5}
Exercise 68, problem 4, p. 129
The sum of the two digits of a number is $9$, and if the number is divided by the sum of its digits, the quotient is $6$. Find the number.
Printed answer:
- $54$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {N: 54, x: 5, y: 4}

Exercise 69

Exercise 69, problem 1, p. 130
A sum of money, at simple interest, amounted in $5$ years to $$3000$, and in $6$ years to $$3100$. Find the sum and the rate of interest.
Printed answer:
- $$2500$ at $4$%.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 2500, y: 4}
Exercise 69, problem 2, p. 130
A sum of money, at simple interest, amounted in $10$ months to $$1680$, and in $18$ months to $$1744$. Find the sum and the rate of interest.
Printed answer:
- $$1600$ at $6$%.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 1600, y: 6}
Exercise 69, problem 3, p. 130
A man has $$10,000$ invested, a part at $4$%, and the remainder at $5$%. The annual income from his $4$% investment is $$40$ more than from his $5$% investment. Find the sum invested at $4$% and at $5$%.
Printed answer:
- $$6000$ at $4$%; $$4000$ at $5$%.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 6000, y: 4000}
Exercise 69, problem None, p. 130
A sum of money, at simple interest, amounted to $$2480$ in $4$ years, and to $$2600$ in $5$ years. Find the sum and the rate of interest.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles

Exercise 70

Exercise 70, problem 1, p. 131
Half the sum of two numbers is $20$; and $5$ times their difference is $20$. Find the numbers.
Printed answer:
- $22$ and $18$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": 22, "y": 18}
Exercise 70, problem 2, p. 131
A certain number when divided by a second number gives $7$ for a quotient and $4$ for a remainder. If three times the first number is divided by twice the second number, the quotient is $11$ and the remainder $4$. Find the numbers.
Printed answer:
- $60$ and $8$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": 60, "y": 8}
Exercise 70, problem 3, p. 131
A fraction becomes $\frac{4}{5}$ in value by the addition of $2$ to its numerator and $3$ to its denominator. If $2$ is subtracted from its numerator and $1$ from its denominator, the value of the fraction is $\frac{3}{4}$. Find the fraction.
Printed answer:
- $\frac{14}{17}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": 14, "y": 17}
Exercise 70, problem 4, p. 131
A farmer sold $50$ bushels of wheat and $30$ of barley for $74$ dollars; and at the same prices he sold $30$ bushels of wheat and $50$ bushels of barley for $70$ dollars. What was the price of the wheat and of the barley per bushel?
Printed answer:
- Wheat, $$1$; barley, $$\frac{4}{5}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"b": "4/5", "w": 1}
Exercise 70, problem 5, p. 131
If A gave $$10$ to B, he would then have three times as much money as B; but if B gave $$5$ to A, A would have four times as much as B. How much has each?
Printed answer:
- A, $$235$; B, $$65$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"A": 235, "B": 65}
Exercise 70, problem 6, p. 131
A and B have together $$100$. If A were to spend one-half of his money, and B one-third of his, they would then have only $$55$ between them. How much money has each?
Printed answer:
- A, $$70$; B, $$30$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"A": 70, "B": 30}
Exercise 70, problem 7, p. 131
A fruit-dealer sold $6$ lemons and $3$ oranges for $21$ cents, and $3$ lemons and $8$ oranges for $30$ cents. What was the price of each?
Printed answer:
- Lemon, $2$ cts.; orange, $3$ cts.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"L": 2, "O": 3}
Exercise 70, problem 8, p. 131
If A gives me $10$ apples, he will have just twice as many as B. If he gives the $10$ apples to $B$ instead of to me, A and B will each have the same number. How many apples has each?
Printed answer:
- A, $30$ apples; B, $10$ apples.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"A": 30, "B": 10}