Envision Math Grade 8 Volume 1 Chapter 2 Analyze And Solve Linear Equations
Page 103 Exercise 1 Answer
Given:
Jasmine’s expression: 2(3x + 6)
James’s expression: 3(2x + 4)
We consider the table with four more values and draw conclusion:
We observe that whichever number we take, the result is same for both the expressions. This proves that Jasmine and James are of the same age. Since Jasmine and James are twins, this table yields the same result. This is true for every whole number.
We observe that whichever number we take, the result is same for both the expressions. This proves that Jasmine and James are of the same age.
Since Jasmine and James are twins, this table yields the same result. This is true for every whole number.
Page 104 Question 1 Answer
A one-variable equation is the equation which only has one variable.
If the equation two or more variables then it becomes a linear equation in two variables or so on.
The solution of an equation is based on the number of variable present in the equation.
Therefore, one-variable equations will always have on unique solution.
Yes, a one variable equation always have one and unique solution.
Page 104 Exercise 1 Answer
There are two ways to solve this problem.
The first way is to draw a bar diagram to represent the perimeters. Then we have to decompose and reorder the bar diagram to solve for x.
The other way is to write an equation to represent equal perimeters. Then use inverse operations and properties of equality to solve.
Then draw the bar diagram.
Choose whichever way is easier.
We can use bar diagram to represent the equal perimeters by first drawing a bar diagram to represent the perimeters.
Then we have to decompose and reorder the bar diagram to solve for x.
The other way is to write an equation to represent equal perimeters. Then use inverse operations and properties of equality to solve for x.
Then draw the bar diagram.
Page 105 Exercise 2 Answer
Given:
The equation is:
Because 8≠3 there is no solution for the given expression.
Page 106 Exercise 3 Answer
The equation is:
3x + 1.5 = 2.5x + 4.7
When we mentally solve this equation we get x = 6.4
Therefore the given equation 3x + 1.5 = 2.5x + 4.7 has only one solution.
Therefore the equation 3x + 1.5 = 2.5x + 4.7 has one solution.
Given:
The equation is:
3(x + 2) = 3x − 6
When we mentally solve this equation we get 6 ≠ −6
Therefore the given equation 3(x + 2) = 3x − 6 has no solution.
Therefore the equation 3(x + 2) = 3x − 6 has no solution.
Given:
The equation is:
9x − 4 = 5x − 4 + 4x
When we mentally solve this equation we get 9x − 4 = 9x − 4
Therefore the given equation 9x − 4 = 5x − 4 + 4x has infinitely many solutions.
Therefore the equation 9x − 4 = 5x − 4 + 4x has infinitely many solutions.
Page 104 Exercise 1 Answer
When we solve an equation, we generally obtain a value of x.
The value of x obtained by solving the equation is the solution of the equation.
There is a possibility that the solution obtained is a whole number, rational number, a fraction or an integer. Nonetheless, it is the solution of the equation.
So yes, if the value of x is negative, the equation will still be true.
Yes, if the value of x is negative, the equation will still be true because, the value of x negative or positive is still the solution of the equation.
Page 107 Exercise 1 Answer
A one-variable equation is an equation that only has one variable.
If the equation two or more variables then it becomes a linear equation in two variables or so on.
The solution of an equation is based on the number of variables presents in the equation.
Therefore, one-variable equations will always have on a unique solution.
Yes, a one-variable equation always have one and unique solution.
Page 107 Exercise 2 Answer
Given:
The equation is:
6x + 12 = 2(3x + 6)
When we mentally solve this equation we get 6x + 12 = 6x + 12
Therefore the given equation 6x + 12 = 2(3x + 6) has infinitely many solutions.
Therefore Kaylee’s equation 6x + 12 = 2(3x + 6) has infinitely many solutions.
Page 107 Exercise 3 Answer
Given:
Height of the first plant is represented by the expression: 3(4x + 2)
Height of the second plant is represented by expression: 6(2x + 2)
We consider the two expressions and put them into a table to see if for the same whole, they yield the same result.
We observe that even after days the plants do not grow of the same height.
No, it is not possible for the plants to be of the same height.
Page 107 Exercise 4 Answer
Given:
The equation is: 3(2.4x + 4) = 4.1x + 7 + 3.1x
To find : solve the given equation
We consider:
Because 12 ≠ 7 the equation has no solution.
The equation 3(2.4x + 4) = 4.1x + 7 + 3.1x has no solution.
Page 107 Exercise 5 Answer
Given:
The equation is:
7x + 3x − 8 = 2(5x − 4)
We consider:
7x + 3x − 8 = 2(5x − 4)
10x − 8 = 10x − 8
Because 10x − 8 = 10x − 8 the equation has infinitely many solutions.
The equation 7x + 3x − 8 = 2(5x − 4) has infinitely many solutions.
Page 107 Exercise 6 Answer
Given:
Todd buys peaches and a carton of vanilla yogurt. Agnes buys apples and a jar of honey
They bought the same number of pieces of fruit.
Peaches = $1.25 each
Vanilla Yogurt = $4
Apples = $1 each
Honey = $6
Let x be the number of fruits bought by Todd and Agnes.
Forming the two equations
Todd:
1.25x + 4
Agnes:
1x + 6
We equate the two equations:
1.25x + 4 = 1x + 6
0.25x = 2
x = 8
If both Agnes and Todd buy 8 fruits, then it is possible that they both pay the same amount.
The situation in which Agnes and Todd pay the same amount for their purchases is if they buy 8 fruits each.
Page 108 Exercise 8 Answer
Given:
The given equation is 4(4x + 3) = 19x + 9 − 3x + 3
To find : solve the given equation
We consider:|
Since 12 is equal to 12, the equation has infinite solutions.
Since, the equation 4(4x + 3) = 19x + 9 − 3x + 3 has infinite solutions.
Page 108 Exercise 11 Answer
Given:
Store A’s prices are represented by the expression 15x − 2
Store B’s prices are represented by the expression 3(5x + 7)
Let x be the rates.
Equating the two equations
We consider:
15x − 2 = 3(5x + 7)
15x − 2 = 15x + 21
−2 ≠ 21
Since −2 ≠ 21 the store never charges the same rate.
We observe that −2 ≠ 21, therefore, the store never charges the same rate.
Page 109 Exercise 12 Answer
When the equation is equivalent to 0 = 0 the given equation will have infinitely many solutions.
When the equation is equivalent to a ≠ b, a and b being the two solutions, the given equation will not have any solution.
The equations having infinite solutions or no solutions will keep on going no matter how many times we get no solution and no matter how many times we get an infinite number.
Solving equations with no solution are similar to solving equations with infinite solutions because both will keep on going no matter how many times
we get no solution and no matter how many times we get an infinite number.
Page 109 Exercise 13 Answer
Given:
The given equation is: 0.9x + 5.1x − 7 = 2(2.5x − 3)
To find: solve the given equation
We consider:
0.9x + 5.1x − 7 = 2(2.5x − 3)
6x − 7 = 5x − 6
x = 1
The equation has only one solution.
The equation 0.9x + 5.1x − 7 = 2(2.5x − 3) has only one solution.
Page 109 Exercise 15 Answer
Given:
The given equation is: 49x + 9 = 49x + 83
We consider:
49x + 9 = 49x + 83
49x − 49x + 9 = 49x − 49x + 83
9 ≠ 83
The equation does not have any solution.
The equation 49x + 9 = 49x + 83 has no solution.
The given equation is: 49x + 9 = 49x + 83
To find: solve the given equation
Solution:
49x + 9 = 49x + 83
+9 = 83 which is false. So, the equation has no solution.
Examples of equations having no solution is:
−9(x + 6) = −9x + 108 and 7(y − 8) = 7y + 42
−9(x + 6) = −9x + 108 and 7(y − 8) = 7y + 42 are the equations in one variable that have no solutions.
Page 109 Exercise 16 Answer
The given equation is: 6(x + 2) = 5(x + 7)
The given equation 6(x + 2) = 5(x + 7) has only one solution.
Page 109 Exercise 17 Answer
The given equation is: 6x + 14x + 5 = 5(4x + 1)
To find: Write a word problem or any expression that this expression represents
The equivalent form of 6x + 14x + 5 = 5(4x + 1) is 100x + 25 = 100x + 25
The given equation 6x + 14x + 5 = 5(4x + 1) has infinite many solutions.
Page 110 Exercise 19 Answer
The equations should have one equation, no solution and infinite many solutions.
The equation that have one solution:
2x + 1 = 9
The equation that has no solution:
5x − 3x + 6 = 2x + 7 − 2
The equation that have infinitely many solutions:
7(8x + 5) − 35 = 4(14x)
2x + 1 = 9 have only one solution
5x − 3x + 6 = 2x + 7 − 2 has no solution
7(8x + 5) − 35 = 4(14x) has infinite many solution.
Page 110 Exercise 20 Answer
The given equation is: 4(4x − 2) + 1 = 16x − 7
The given equation has no solution.
Equation 4(4x – 2) + 1 = 16x – 7 has no solution.
Page 110 Exercise 21 Answer
The given equation is : 6x + 26x − 10 = 8(4x + 10)
To find: solve the given equation
The value of x is 15.
Page 110 Exercise 22 Answer
The given equation is 64x − 16 = 16(4x − 1)
The given equation 64x – 16 = 16(4x – 1) has infinite many solutions.
Page 110 Exercise 23 Answer
The given equation is 5(2x + 3) = 3(3x + 12)
To find: solve the given equation
The given solution 5(2x + 3) = 3(3x + 12) has only one solution.
Page 110 Exercise 24 Answer
The given equation is: 4(2x + 3) = 16x + 12 − 8x
To find: Which of the following best describes the solution to the equation
The given equation 4(2x + 3) = 16x + 12 − 8x has infinite many solutions.
Page 110 Exercise 25 Answer
The given equation is: 10x + 45x − 13 = 11(5x + 6)
To find: solve the given equation
Which is false, so the equation has no solution.
The statement which is true is: the equation has no solution.