Solution:
Uphill x flat road y downhill z
x+y+z = 142 x/28+y/30+z/35 = 4.5 z/28+y/30+x/35 = 4.7
Answer: x=42 y=30 z=70.
2. 65 1 junior high school students in a school. Senior two is more than senior three 10%, and senior one is more than senior two by 5%. How many people are there in each of the three grades?
Solution:
Grade one: X Grade two: Y Grade three: Z.
x+y+z = 65 1y = 1. 1z x = 1.05y
Answer: x=23 1 y=220 z=200.
3.x+y= 10
2x-3y+2z=5
x+2y-z=3
Solution:
x+y= 10 - ( 1)
2x-3y+2z=5 - (2)
x+2y-z=3 - (3)
(3)*2+(2)
4x+y= 1 1 - (4)
(4)-( 1)
3x= 1
x= 1/3
Substituting x= 1/3 into (1), the result is
y=29/3
Substitute x= 1/3 and y=29/3 into (3) to get the result.
z=50/3
4. 65 1 person in three grades of junior high school in a school. Senior two is more than senior three 10%, and senior one is more than senior two by 5%. How many students are there in each of these three grades?
Solution:
At the beginning of the solution,12 3 people were X Y Z respectively.
X+Y+Z=65 1
Y= 1 10%Z
X= 105%Y
(in the process of solving, you must change to z to operate. )
23 1/200 Z+220/200 Z+200/200 Z = 65 1
Z=200 Y=220 X=23 1
5. In the square of the algebraic expression ax +bx+c, when x = 1, 2 and -3, the values of the algebraic expression are 0, 3 and 28 respectively. What kind of algebraic expression is this?
Solution:
According to the meaning of the problem, the equation is obtained:
A+b+c=0 Equation 1
4a+2b+c=3 Equation 2
9a-3b+c=28 Equation 3
Equation 2- Equation 1, so:
3a+b=3
Equation 3- Equation 1, so:
5a-5b=25, that is, a-b=5.
Get a new set of equations:
3a+b=3
a-b=5
Solve the equation:
a=2
b=-3
Substituting A = 2 and B =-3 into the original equation, we get c= 1.
So the solutions of the original equations are: a=2, b=-3, c= 1.
6。 The square of y = a * x+bx+c in the equation, when x= 1, y =-2; When x=- 1, y = 20. When x=3/2 and x= 1/3, the values of y are equal. Find the values of a, b and c.
Solution: when x= 1, y =-2; When x=- 1 and y=20, the equations 1.2 are listed respectively.
a+b+c=-2............. 1
a-b+c=20.............2
a+b=0.............3
So b =-11a =11c =-2.
7.36 bricks, 36 people moved, 4 men and 3 women moved, and two children moved one. How many men, women and children are there?
Solution:
Let there be a man, a woman and a child. According to the meaning of the question, we can get the system of equations.
4a+3b+0.5c=36，
a+b+c=36。
Find the integer solution of this equation,
Eliminate c to get 7a+5b=36.
7a can only take 7, 14, 2 1, 28,
5b can only take 5, 10, 15, 20, 25,
Of these figures, only 2 1+ 15 = 36, and others are not.
At this time, a=3, b=3 and c=30.
There are 3 males, 3 females and 30 children.
8. A three-digit number, the sum of unit and percentile is equal to ten digits, seven times of percentile is two times larger than unit and percentile, and the sum of unit and percentile is 14. Find this three-digit number
Solution:
Let one digit = X, ten digits = Y, and hundred digits = z.
Yes: x+z = y .................. (1)
7z = x+y+2 …………( 2)
x+y+z = 14 …………( 3)
In order to solve this system of equations, check (2), there are:
x + y = 7z - 2
Substituting into (3), there are
8z = 16
So: z = 2
The results are: y = 7, x = 5.
This three-digit number = 275
9. Let y+z/x=x+y/z=x+z/y=k, and find k?
Solution:
Y+Z=XK
X+Y=ZK
Z+X=YK
2(X+Y+Z)=K(X+Y+Z)
K=2
10. 100 yuan to buy 100 chickens, including 5 cocks, 3 hens and 3 chicks. Ask how many cocks, hens and chickens there are.
Solution: Solution: Set X cocks, Y hens and Z chickens.
You can get it according to the meaning of the question.
x+y+z= 100
5x+3y+z/3= 100
Get after reduction
7x+4y= 100
Looking at the equation, we can see that 25-7x/4 must be an integer.
X is 4,8, 12.
If x=4, then y= 18, then z=78.
If x=8, then y= 1 1, then z=8 1.
If x= 12, then y=4, then z=84.
Review exercises of binary linear equations
Fill in the blanks
1, the equation about x is a linear equation when _ _ _ _ _ _ _ _; When _ _ _ _ _ _ _ _ _, it is a binary linear equation.
2. It is known that the formula expressed by is _ _ _ _ _ _ _ _ _ _ _; The formula represented by is _ _ _ _ _ _ _. When _ _ _ _ _ _ _ _; Write its two positive integer solutions _ _ _ _ _ _ _.
3. If the equation 2x+y = is a binary linear equation, then mn=.
4, known and have the same solution, then = _ _, =.
5, if known, the value is.
6. If so _ _ _ _ _.
7. If (x-y) 2+| 5x-7y-2 | = 0, then x = _ _ _ _ _ _ _ and y = _ _ _ _ _ _ _
8. It is known that Y = KX+B, if X = 4, y =15; When x = 7 and y = 24, then k =；; b=。
9, called a solution of the equation, then.
10, the positive integer solution of the binary linear equation 4x+y=20 is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
1 1. Take out 5 points from coins with 1, 2 and 5 points, and the method of * * * is the same as _ _ _ _ _ _ (in no particular order).
12, the solution of the equations is _ _ _ _ _ _ _ _ _ _.
13. If the solution of binary linear equations is 0, then A+B = _ _ _ _ _ _.
14, the solution of the equation is
15, given 6x-3y = 16, 5x+3y = 6, and 4x-3y has the following values.
16, if it is the solution of the equation about sum, then =.
17. It is known that the waist center line of an isosceles triangle divides its circumference into 63 and 36 parts, so its waist length is _ _ _ _ _ _ _. The length of the bottom is _ _ _ _ _.
18, it is known that point A (-y- 15,-15-2x) and point B (3x, 9y) are symmetrical about the origin, so the value of x is _ _ _ _ _ _ _ _ _ _.
Second, multiple choice questions.
1, in the system of equations,,,, is a binary linear system of equations ().
a，2 B，3 C，4 D，5。
2, the solution of binary linear equations is ()
A.B. C. D。
3. Three binary linear equations 2x+5y-6 = 0, 3x-2y-9 = 0, y = kx-9 have a common * * * solution if k= ().
A.4 B.3 C.2 D. 1
4. As shown in the figure, eight identical small rectangular floor tiles are spliced into a rectangle, and the area of each small rectangle is ().
A.400 square centimeters B. 500 square centimeters C. 600 square centimeters D. 675 square centimeters
5. The price of a coke is 1.8 yuan. In order to promote sales, customers get a lottery ticket for every bottle of coke they buy, and every three lottery tickets can be exchanged for a bottle of coke, so each lottery ticket is equivalent to ().
(1) 0.6 yuan (2) 0.5 yuan (3) 0.45 yuan (4) 0.3 yuan.
6, known the solution of the equation, then, the relationship between, is ()
A, B, C, D,
7. In order to protect the ecological environment, a county in Shaanxi province responded to the call of "returning farmland to forests" and changed part of cultivated land into forest land. After the change, the forestland area and cultivated land area * * * are 180 square kilometers, and the cultivated land area is 25% of the forestland area. In order to find out how many square kilometers the forestland area and cultivated land area are after the change. Let the changed cultivated land area be x square kilometers and the forestland area be y square kilometers. According to the meaning of the question, list the following four equations, of which the correct one is ().
A B C D
8. Suppose the distance between Town A and Town B is 1000 meters. A starts from town A and town B at the same time and travels in the opposite direction. The speeds of A and B are 1000 m/h and 1000 m/h respectively, and they meet 30 minutes after departure. (2) A immediately returned to town B, and it took another 30 minutes to catch up with B; When A caught up with B, they were still 4 kilometers away from a town. Seek … According to the meaning of the question and condition ③, each of the four students draws the following third equation, and the wrong one is ().
A, B, C, D,
Third, answer the question.
1, in y=, when the value of y is, when the value of y is, when the value of y is, when the value of y is, find the value of y.
There are three kinds of stairs: five-step ladder, seven-step ladder and nine-step ladder, and the height of each step is the same. The handrail length (ladder length), top width and bottom width of each staircase are shown in the figure, and the tenon joint between the rung and the handrail is called the connection point (such as point A).
(1) Fill in the following table by calculation:
stairway
Total length of two handrails (m) Total length of crosspiece (m) Connection point (unit)
Five-step ladder 4 2.0 10
Seven steps
Nine steps
(2) The cost of stairs consists of material cost and processing cost. Assuming that the processing cost is calculated at each connection point 1 yuan, the unit price of handrails in the material cost is not equal to the unit price of rungs (ignoring factors such as material loss). Now we know that the costs of the five-step ladder and the seven-step ladder are 26 yuan and 36 yuan respectively. Try to find out the cost of the nine steps.
3. Solve the following equation
( 1) ⑵
4. In League A and League B, according to the scoring rules of the Football Association, a football team and its reward scheme are as follows.
Win a game, draw a game, lose a game.
Integer 3 1 0
Bonus (RMB/person) 1500 700 0
By the end of round 12, the team had lost three games, and the score was 19.
Q: (1) How many games did the team win and draw? (2) If each player gets the 500 yuan appearance fee in each game, try to get the total income of each player after the 12 round.
The reference answer is as follows:
Solution: (1) The handrails of the seven-step ladder and the nine-step ladder are 5m and 6m respectively. The total length of crosspiece is 3.5m and 3.5m respectively (each 1 min); The number of connection points is 14 and 18 respectively.
(2) Set the unit price of handrails as X yuan/m, and the unit price of crosspieces as Y yuan/m ... According to the meaning of the question:
Which is the solution. So the cost of the nine-step ladder is 6× 3+5.4× 2+/kloc-0 /×18 = 46.8 yuan (9/).
A: The cost of a nine-step ladder is 46.8 yuan.
Respondent: 45249 1860- probation period level 1 8-211:08.
... some trouble
Respondent: bumin 03 12- elementary 8-23 20:54
1. Binary linear equation 4x-3y= 12. When x = 0, 1, 2, 3, y = _ _ _ _ _
2. In x+3y=3, if X represents Y, then Y = _ _ _ _ _ and Y represents X, then X = _ _ _ _ _
4. The general form of transforming Equation 3(x+5)=5(y- 1)+3 into a binary linear equation is _ _ _ _.
(1) Equation y=2x-3 has _ _ _ _ _ _;
(2) The solution of equation 3x+2y= 1 has _ _ _ _ _ _;
(3) The common * * solution of equations y=2x-3 and 3x+2y= 1 is _ _ _ _ _.
9. Equation x+y=3 has _ _ _ group solutions and _ _ _ group positive integer solutions, which are _ _ _ _ _ respectively.
1 1. Equation (k2-1) x2+(k+1) x+(k-7) y = k+2. When k = _ _ _ _ _ _, the equation is linear; When k = _ _ _ _ _, the equation is a binary linear equation.
12. For the binary linear equation 2(5-x)-3(y-2)= 10, when x=0, then y = _ _ _ _ _ _ When y=0, then x = _ _ _ _.
13. The positive integer solution of equation 2x+y=5 is _ _ _ _ _.
14. If (4x-3)2+|2y+ 1|=0, then x+2 = _ _ _ _.
Solution.
When k is _ _ _ _ _, the equations have no solution.
______.
(2) Choose
24. in equation 2(x+y)-3(y-x)=3, if y is expressed by an algebraic expression containing x, then []
a . y = 5x-3；
b . y =-x-3；
D.y=-5x-3。
[ ]
26. The system of equations consisting of the known binary linear equation 5x-y=2 has many solutions [].
a . 10x+2y = 4；
b . 4x-y = 7；
c . 20x-4y = 3；
D. 15x-3y=6。
[ ]
Morning = 9;
b . m = 6；
c . m =-6；
D.m=-9。
28. If 5x2ym and 4xn+m- 1y are similar terms, the value of m2-n is [].
a . 1；
B.- 1;
C.-3;
D. None of the above answers are correct.
29. The solution of equation 2x+y=9 in the positive integer range is [].
a . 1；
B.2;
C.3;
D. iv
[ ]
A.4
B.2
C.-4;
D. None of the above answers are correct.
Binary linear equations? Comprehensive innovation exercise
First, comprehensive questions
z，3，2
z，3，2
3. It is known that 4ax+yb2 and-a3by are 2x-y values of similar terms.
z，3，2
4. If | x-2 |+(2x-3y+5) 2 = 0, find the values of x and y. 。
Three, three
5. If the equation 2x2m+3+3Y5n-4 = 7 is a binary linear system of x and y, find the value of m2+n. 。
z，3，2
Second, the topic of innovation
1. It is known that x and y are reciprocal, and (X+Y+4) (X-Y) = 4. Find the values of x and y.
n，4，3
2. Find the solution of equation X+2Y = 7 in the range of natural numbers.
n，4，3
Third, the mid-term exam questions
(Shandong, 95, 3 points) The following conclusion is correct.
[ ]
Reference answers and instructions
Knowledge points of 1. 1. test: the solution of equation and the value of algebraic expression.
∴ 2m+3n=2×2+3(-3)=4-9=-5。
2. Knowledge points of the test: the solution of the equation and the solution of the linear equation of one variable.
Solution: Substitute X =-3 and Y =-2 into the equation and get 2 (-3)-4 (-2)+2A x=-3.
Hugging: The knowledge points of the above two questions are similar. When the solution of the equation is known, the values of other letters in the equation can be obtained by substituting this group of numbers into the equation or equation group.
3. Knowledge points of the test: similarity terms and solution equations
Cuddle: According to the definition of similar terms, the same letters have the same index, so the equation can be listed and solved.
4. Knowledge points of the test: the nature of non-negative numbers and the simple solution of binary linear equations.
Cuddling: Because | x-2 | ≥ 0 and (2x-3y+5) 2 ≥ 0, when their sum is zero, the two numbers must be zero, that is, x-2 = 0 and 2x-3y+5 = 0.
5. Knowledge point: the definition of binary linear equation.
Solution: from the meaning of the question.
Hugging: According to the definition of binary linear equation, the exponent of unknown term is 1, from which it is concluded that 2m+3 = 1, 5n-4 = 1.
2. 1. Knowledge points of the exam: the meaning of reciprocal and the solution of simple binary linear equations.
Solution: from the meaning of the question, x+y = 0,
∫(x+y+4)(x-y)= 4。
∴ 4(x-y)=4
X-y = 1。
2. Knowledge points of the test: natural number solution of binary linear equation.
Solution: Transform the equation X+2Y = 7 to get X = 7-2Y.
Let y= 1, 2,3,4 ..., then x = 5,3, 1,-1 ...
Cuddle: Natural number solutions of binary linear equations, that is, unknowns, are all natural numbers. Firstly, the equation is deformed, and another letter is represented by an algebraic expression containing one letter, and then it is solved according to the characteristics of the topic.
Third, knowledge points: the definition of the solution of binary linear equations.
Solution: d
Hugging: according to the definition of binary linear equations, the solution of binary linear equations is the solution of every binary linear equation in the equations, so choose D.