) b−4=−3 b= y3=−12 y= −2(r+4)=36 r= a2−4+a4=14 a= 2+13t=1+14t t= 8(h−1)=6h+4+2h h=
To solve the equations given:
1) b - 4 = -3
Adding 4 to both sides gives:
b = 1
2) y/3 = -12
Multiplying both sides by 3 gives:
y = -36
3) -2(r + 4) = 36
Expanding and simplifying gives:
-2r - 8 = 36
Adding 8 to both sides gives:
-2r = 44
Dividing both sides by -2 gives:
r = -22
4) a^2 - 4 + a^4 = 14
Rearranging the terms gives:
a^4 + a^2 - 18 = 0
This can be factored as:
(a^2 - 3)(a^2 + 6) = 0
Setting each factor equal to zero gives:
a^2 - 3 = 0 or a^2 + 6 = 0
Solving the first equation gives:
a^2 = 3
Taking the square root of both sides gives:
a = ±√3
Solving the second equation gives:
a^2 = -6
But since square roots of negative numbers are not real, there are no solutions for this equation.
So the possible solutions for a are:
a = √3 or a = -√3
5) 2 + 13t = 1 + 14t
Subtracting 1 and 13t from both sides gives:
1 = 14t - 13t
Simplifying gives:
1 = t
6) 8(h - 1) = 6h + 4 + 2h
Expanding and simplifying gives:
8h - 8 = 8h + 4
Subtracting 8h from both sides gives:
-8 = 4
This is not a true statement, so there is no solution for this equation.
Let's solve the equations step by step:
1) b - 4 = -3
To solve for b, we need to isolate it. Add 4 to both sides:
b - 4 + 4 = -3 + 4
b = 1
2) y/3 = -12
To solve for y, multiply both sides by 3:
(y/3) * 3 = -12 * 3
y = -36
3) -2(r+4) = 36
First, distribute the -2:
-2 * r + (-2) * 4 = 36
-2r - 8 = 36
Next, add 8 to both sides:
-2r - 8 + 8 = 36 + 8
-2r = 44
Divide both sides by -2:
(-2r)/(-2) = 44/(-2)
r = -22
4) a^2 - 4 + a^4 = 14
Rearrange the equation:
a^4 + a^2 - 4 = 14
Combine like terms:
a^4 + a^2 - 4 - 14 = 0
a^4 + a^2 - 18 = 0
This is a quadratic equation in terms of a^2. Let's say x = a^2:
x^2 + x - 18 = 0
Factor the quadratic equation:
(x - 3)(x + 6) = 0
This gives us two possible values for x:
x - 3 = 0 or x + 6 = 0
x = 3 or x = -6
Since x = a^2, we can take the square root of both sides to find a:
a = √3 or a = √(-6)
The square root of a negative number is not a real number, so a = √(-6) is not a valid solution. Therefore, a = √3.
5) 2 + 13t = 1 + 14t
To solve for t, we need to isolate it. Subtract 1 from both sides:
2 - 1 + 13t = 1 - 1 + 14t
1 + 13t = 14t
Subtract 13t from both sides:
1 + 13t - 13t = 14t - 13t
1 = t
6) 8(h - 1) = 6h + 4 + 2h
First, distribute the 8 on the left side of the equation:
8h - 8 = 6h + 4 + 2h
Combine like terms:
8h - 8 = 8h + 4
Subtract 8h from both sides:
8h - 8 - 8h = 8h + 4 - 8h
- 8 = 4
Since this equation is not true (-8 does not equal 4), there is no valid solution for h.
Let's break down each equation step by step to find the value of the variable in each equation.
1) b - 4 = -3
To isolate the variable b, we need to move the constant term (-4) to the other side of the equation by adding 4 to both sides:
b - 4 + 4 = -3 + 4
This simplifies to:
b = 1
Therefore, the value of b is 1.
2) y/3 = -12
To isolate the variable y, we can multiply both sides of the equation by 3:
3 * (y/3) = -12 * 3
Simplifying further:
y = -36
So, the value of y is -36.
3) -2(r + 4) = 36
We will simplify this equation step by step. First, distribute -2 to the terms inside the brackets:
-2 * r - 2 * 4 = 36
This simplifies to:
-2r - 8 = 36
Next, we will move the constant term (-8) to the other side of the equation by adding 8 to both sides:
-2r - 8 + 8 = 36 + 8
This simplifies to:
-2r = 44
To solve for r, we divide both sides of the equation by -2:
-2r / -2 = 44 / -2
This gives us:
r = -22
Therefore, the value of r is -22.
4) a^2 - 4 + a^4 = 14
We will simplify this equation step by step. First, let's combine like terms on the left side:
a^4 + a^2 - 4 = 14
Next, move the constant term (-4) to the other side by adding 4 to both sides of the equation:
a^4 + a^2 - 4 + 4 = 14 + 4
This simplifies to:
a^4 + a^2 = 18
To solve for a, we can factor out a^2 from both terms on the left side:
a^2(a^2 + 1) = 18
Now, divide both sides of the equation by a^2 + 1:
(a^2(a^2 + 1))/(a^2 + 1) = 18/(a^2 + 1)
This gives us:
a^2 = 18/(a^2 + 1)
To remove the square on the left side, we take the square root of both sides:
√(a^2) = √(18/(a^2 + 1))
This simplifies to:
a = ±√(18/(a^2 + 1))
The value of a can be either positive or negative square root of 18 divided by (a^2 + 1).
5) 2 + 13t = 1 + 14t
We will isolate the variable t step by step. First, let's move the constant term (1) to the other side by subtracting 1 from both sides:
2 + 13t - 1 = 1 + 14t - 1
This simplifies to:
13t + 1 = 14t
Next, let's move the variable terms to one side and the constant term to the other side:
13t - 14t = -1
Simplifying further:
-t = -1
To solve for t, we multiply both sides of the equation by -1:
-t * -1 = -1 * -1
This gives us:
t = 1
Therefore, the value of t is 1.
6) 8(h - 1) = 6h + 4 + 2h
We will simplify this equation step by step. First, let's distribute the 8 to the terms inside the brackets:
8h - 8 = 6h + 4 + 2h
Next, let's combine like terms on both sides of the equation:
8h - 8 = 8h + 4
To isolate the variable h, we can subtract 8h from both sides of the equation:
8h - 8 - 8h = 8h + 4 - 8h
Simplifying further:
-8 = 4
This is not a true statement, which means there is no value of h that satisfies the equation. Therefore, there is no solution for h in this equation.
The formula for the area of a triangle is A = bh/2 .(2 points) Solve the formula for height. h = A triangle has a base of 7cm and an area of 28 sq cm. What is its height? cm
To solve the formula for height, we need to isolate the variable h.
A = (bh) / 2
Multiply both sides of the equation by 2:
2A = bh
Divide both sides of the equation by b:
(2A) / b = h
So the formula for height is:
h = (2A) / b
Now, let's find the height of a triangle with a base of 7cm and an area of 28 sq cm.
Given:
b = 7 cm
A = 28 sq cm
Plugging these values into the formula for height, we have:
h = (2 * 28) / 7
Simplifying further:
h = 56 / 7
h = 8 cm
Therefore, the height of the triangle is 8 cm.