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The four points can be plotted:
.

.
The diagonals are defined by the points: U,W and V,X.
.
We need to solve for a line going through the first pair of points
U (-2,1)
W (4,-1)
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y = mx + b
m= (-1-1)/(4-(-2))= -2/6 = -1/3
y = -1/3x + b
We can solve for 'b' by plugging in the points we know.
1 = -1/3*(-2 ) + b
b = 1/3
or
-1 = -1/3*4 +b
b = 1/3
.
y=-1/3x + 1/3 defines the equation for the first diagonal.
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Then we need to find the equation for the line going through the other two points
V (2,3)
X (0,-3)
m= (-3-3)/(0-2) = -6/-2 = 6/2 = 3
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At this point we know the diagonals are perpendicular because, by definition, the slopes of perpendicular lines are the negative reciprocal.
-1/3 is the negative reciprocal of 3.
.
But we can press on to continue to define the second equation...
.
-3 = 3(0) + b
b = -3
or
3 = 3(2) + b
b = -3
y = 3x -3
.
Then a graph will illustrate they are perpendicular.
.

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With solution problems, you need to keep track of the amount of 'pure' stuff.
You need to end up with 120 ml of a 20% solution, so that is
.20 * 120 = 24 ml of 'pure' acid
120-24 = 96 ml of water
.
x = volume of 10% acid
y = volume of 50% acid
.
x+y = 120 ml
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It is better to set up the equation with the minimum number of unknowns.
In this case, we know the 120 ml is required, so we can say:
x = volume of 10% acid
120-x = volume of 50% acid
.
.1*x + .5(120-x) = .2*120
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multiply both sides of the equation by 10 to get rid of decimals
.
10*(.1*x + .5(120-x)) = 10*(.2*120)
10*(.1*x) + 10(.5(120-x)) = 10*(.2*120)
x + 5(120-x) = 2*120
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x + 600 -5x = 240
.
-4x = -360
.
x = 90
.
y = 120-90 = 30
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Checking, how much 'pure' stuff we would have...
.1*90 = 9 ml
.5*30 = 15 ml
9+15 = 24 ml, which is what we needed
.
So the answer is:
90 ml of a 10% solution
plus 30 ml of a 50% solution
to produce 120 ml of 20% solution.
.
Done

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The diagonal of a square is the hypotenuse of a triangle.
.
c^2 = a^2 + b^2
.
Since it is a square, a=b.
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62^2 = a^2 + b^2 = a^2 + a^2
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62^2 = 2a^2
.
3844 = 2a^2
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2a^2 = 3844
.
a^2 = 1922
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a = sqrt(1922)
.
a = 43.84
.
So, the perimeter = 4*43.84 = 175.35 or about 176
.
Done

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No. -5-3=-8.

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Any number that can be written as a fraction is, by definition, a rational number.
.
29.6% is mathematically = 0.296.
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And we see that .296 = 296 thousandths, which we can show as:
.

.

.
The 8's cancel, so we end up with
.

.
Done.

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L = number of buttons collected by Lydia
S = number of buttons collected by Sarah
D = number of buttons collected by Denise
L + S + D = 120 :: Total given.
.
L = S + 20 :: Lydia has 20 more than Sarah
D = 3S :: Denise had 3 times as many as Sarah
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Question: How many did Lydia have?
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We have 3 unknowns, so we need 3 equations...
OR
We need to redefine the variables to use fewer than 3 unknowns.
.
Looking back, the problem was stated in terms of Sarah's buttons.
.
L = S+20
D = 3S
and
S = S of course
.
L + D + S = 120 :: given
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substituting...
.
(S+20) + 3S + S = 120
5S = 100
S = 20
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Substituting this value
.
L = S + 20
L = 20 + 20 = 40
.
D = 3S
D = 3(20) = 60
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Checking, does L+S+D = 120?
20 + 40 +60 = 120
Yes!
.
Rechecking the question, it only asks how many Lydia has.
.
Answer:
Lydia has 40 buttons.
.
Done.

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.
Add 4x to both sides
.

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This is conceptually ax^2 + bx = c
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To complete the square, take 1/2*b, square it, and add it as a new 'c'.
.

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.

.

.

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hmmmm....that means we have an imaginary number in here, since we have a squared term on the left and a negative number on the right.
.

.

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The triangle has 3 angles: a, b, and c.
We know that a + b + c = 180 degrees because all triangles have 180 degrees.
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We'll pick 'a' as the smallest angle.
.
b = a + 23 :: given
.
c = a+b+10 :: given
.
So, it seems we have 3 unknowns and not enough equations to solve them.
.
But let's think about it.
.
We have
a = a
.
b = a+23
.
c = a+b+10
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for 'c', substitute b=a+23
.
c = a + (a+23) +10
.
a + b + c = 180,
so...
a + (a+23) + (a+(a+23)+10) = 180
.
4a + 56 = 180
4a = 124
a = 31
.
b = a +23
b = 31+23 = 54
.
c = a + b + 10
c = 31 + 54 + 10 = 95
.
Do they add up to 180?
31 + 54 + 95 = 180
Correct.
.
The three angles are: 31, 54, and 95.
.
Done.

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.
This is a basic concept you need to take the time to understand.
.
Consider 3/4...

.
Or 1/2...

.
Etc.

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y = x +4 is shown in red.
y = -2x+1 is shown in green.
Graphing...
.

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The point of intersection is (-1,3), which is the single solution.
Since there is at least one solution, we can say the equations are consistent.
The y-intercept of y=x+4 is (0,4)
The x-intercept of y=x+4 is (-4,0)
The slope of y=x+4 = 1, so the line slopes up to the right.
The y-intercept of y=-2x+1 is (0,1).
The x=intercept of y=-2x+1 is (1/2,0).
The slope of y=-2x+1 is -2, so the line slopes down to the right.
.
Done.

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L = length
W = width
A = area
A = L*W
.
L = 2W + 3
.
A = 93 cm^2
.
Substitute L = 2W+3 in the area equation to solve.
.
L*W = 93 cm^2
(2W +3) * W = 93
2W^2 + 3W = 93
2W^2 + 3W -93 = 0
.
This does not factor nicely, so you need to use the quadratic equation.
.

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=753 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 6.11021136700612, -7.61021136700612. Here's your graph:

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This provides two solutions: W=6.11 and W=-7.61.
W cannot be negative, so the approximate value is W=6.11.
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Substituting, W=6.11, we can find L.
L = 93/6.11
L = 15.22 cm
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Checking, does the area = 93 cm^2?
(15.22)(6.11) = 92.9942 cm^2, which is close enough to 93.
Correct.
.
Checking, does L = 2W + 3?
2W = 2*6.11 = 12.22 cm
12.22 + 3 = 15.22 cm
Correct.
.
Answer:
L = 15.22 cm
W = 6.11 cm
.
Done.

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If angles are complementary, then the sum to 90 degrees.
We have two angles, say 'x' and 'y'.
We know from the problem statement they are complementary, so
x+y=90
Given this equation, we can see that it is easier to define them as:
'x' and '90-x'.
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That leaves us with only one unknown to solve.
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In algebra, we'd say:
x = x
y = 90-x
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We also are told one angle is 15 less than twice the other. So if we add 15 to one of the angles, it will equal twice the other angle.
.
y+15 = 2x
.
Which means
y = 2x - 15
.
Returning to the objective equation:
.
x + y = 90
.
Substitute y = 2x-15
.
x + 2x-15 = 90
3x -15 = 90
3x = 105
x = 35
.
x + y = 90
y = 90 -35
y = 55
.
Check to see if these values satisfy the other problem statement:
y + 15 = 2x
y+15 = 55+15 = 70
2x = 2*35 = 70
Correct.
.
Answer:
The two angles are: 35 and 55.
.
Done.

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The minute hand describes a circle as it sweeps across the traditional analog clock face.
The radius of the circle is 7 inches.
The circumference of the circle is the pi*diameter.
The diameter = 2*radius.
C = 2*7*pi
C = 43.98222972 inches
.
I cannot say what your teacher would consider "the most reasonable measure."
Perhaps 44 inches is "the most reasonable measure?"
.
Done.

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A linear equation in standard form is: Ax + By = C
.
We have:
4x -y = 3
.
That is already in standard form.
.
To graph a line, you need two points.
.
I usually pick the y-intercept and x-intercept.
The y-intercept is where x=0.
The x-intercept is where y=0.
.
y-intercept:
4(0) -y = 3
-y = 3
y = -3
(0,-3)
.
x=intercept:
4x -0 = 3
4x = 3
x = 3/4
(3/4,0)
.
Now you can graph it.
.

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Let 'some number' be 'x'.
L = length = x + 8
W = width = x + 8
.
The transitive property teaches that if a=b and b=c, then a=c.
Therefore, L = W.
That means the rug is square.
Of course, a square is an equilateral rectangle, so that is okay.
.
A = area = L*W = L^2 = W^2 = 80 sq. ft.
.
L = sqrt(80) = sqrt(16*5) = sqrt(16)*sqrt(5) = 4*sqrt(5) = 8.94427191
.
So L = W = 8.94427191
.
Returning to the setup, L = x+8 = 8 + .94427191.
So x = .94427191
No wonder he could not remember it.
.
Done.

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B

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We are seeking an equation for a line that will go through the points (4,-3) and (8,-5).
.
We know the slope-intercept form of the equation is: y = mx+b
m = slope
b = y-intercept
.
m = rise / run = change in y / change in x
.
m = (y₂-y₁)/(x₂-x₁)
.
(4,-3) = (x₁,y₁)
(8,-5) = (x₂,y₂)
.
m = (-5 -(-3)) / (8 -4) = (-5 +3) / (8 -4) = -2/4 = -1/2
.
y = -1/2x + b
.
Now we can substitute what we know to find 'b'.
For example, one of the points is (4,-3).
Substituting for x and y...
.
-3 = -1/2(4) + b
-3 = -2 + b
-1 = b
.
And we know the other point is (8,-5)
.
-5 = -1/2(8) + b
-5 = -4 + b
-1 = b
.
b = -1 = y-intercept = (0,-1)
.
y = -1/2x -1
.
Another useful point to find is the x-intercept, which is where y=0.
.
0 = -1/2x -1
1/2x = -1
x = -2
(-2,0)
.
Graph to check
.

.
Answer:
The equation is

.
Done

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Since they have the same denominator, we know that:


.
We notice that 18 > 16, so we can reverse the idea to say:
.


.
Of course, 2/16 can be further simplified by dividing by 2:
.

.
Done

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The reciprocal of x is 1/x.
.


.
'3x' is the common denominator, so multiply through by 3x to cancel the denominators.
.


.
Checking...
.

.


Correct
.
The integer is x=10.
.
Done

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x^2 -8x + 7 = 0 is a quadratic equation.
.
We can factor it.
.
(x-1)(x-7) = 0
.
So x = 1 or x = 7, which are the points (1,0) and (7,0).
.

.

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The given formula is in slope-intercept form: y = mx + b.
y = 3x -9
.
The y-intercept is the point where x=0, so we see it is 'b', which equals -9, or the point (0,-9).
The x-intercept is the point where y=0, so we see it is x=3, or the point (3,0).
With two points you can draw a straight line.
.

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"Two numbers" suggests we use 'x' and 'y'.
However, we are told they sum to 55: x + y = 55
That means we can substitute 55-x for y.
That, in turn, means we have only one unknown to solve.
.
We also are told that x*y = 600
Substituting y = 55-x
.
x*(55-x) = 600
55x -x^2 = 600
.
Add x^2 to both sides
55x = x^2 + 600
.
Subtract 55x from both sides
0 = x^2 -55x + 600
.
Factor
.
(x-15)(x-40)=0
.
So we we have two possible values for x: 15 and 40.
That means we have two possible values for y:
55-15 = 40 and
55-40 = 15.
.
Answer:
The two numbers are 15 and 40. Their sum is 55. Their product is 600.
.
Done.

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This is an interesting application of the Pythagorean Theorem.
The car traveling north can be viewed as going up the y-axis 12 units.
The car going east can be be viewed as going to the right along the x-axis.
It has traveled 'd' miles.
The diagonal from that connects the two cars is the hypotenuse of a right triangle.
It's length is d+8, which is given.
.
12^2 + d^2 = (d+8)^2
144 + d^2 = d^2 +16d + 64
subtract d^2 from both sides
144 = 16d + 64
subtract 64 from both sides
80 = 16d
divide both sides by 16
5 = d
d = 5
.
Check to ensure diagonal 'c' is 5+8 =13.
c^2 = 12^2 + 5^2
c^2 = 144 + 25
c^2 = 169
So, c = 13
Correct.
.
Answer:
The eastbound car has travelled 5 miles.
.
Done.

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You have two equations and two unknowns, so treat them as a system of linear equations:
x+y=12
3x-2y=6
.
multiply the first equation by 2 and add the two equations together
.
2x +2y = 24
3x -2y = 6
5x = 30
x = 6
.
x+y=12
y = 12-x = 12-6 = 6
.
Check by substituting in the second equation.
3x = 3(6) = 18
-2y = -2(6) = -12
18-12 = 6
Correct
.
Answer:
x = 6
y = 6
.
You also can check graphically to see if they two lines cross at (6,6)...
.
Get the equations into slope-intercept form and define the lines.
.
x+y=12
y = -x+12
.
3x-2y=6
-2y = -3x +6
y = 3/2x -3
.

.

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Frank works at a rate of 1/4 job per hr.
James works at a rate of 1/7 job per hr.
It takes them 'x' hours to do one entire job.
x*(1/4) + x*(1/7) = 1
.
x/4 + x/7 = 1
.
7x + 4x = 28
.
11x = 28
.
x = 28/11 = 2.54545
.
check by seeing how much of the job each does in 2.54545 hr
2.54545*1/4 = .6363
2.54545*1/7 = .3636
.6363+.3636 = .9999, which is pretty close to 1 whole job
.
Answer:
Working together it takes them 2.54545 hrs.
Or about 2 hr 33 min
.
Done

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1. Two consecutive integers are best shown as: x and x+1.
(x)(x+1) = 600
x^2 + x -600 = 0
(x+25)(x-24) = 0
x=-25 or x=24
Answer:
The two consecutive integers are either 24 & 25 or -25 & -24.
...
2. Two consecutive even integers are best shown as: x and x+2.
(x)(x+2) = 1848
x^2+2x-1848 = 0
(x+44)(x-42) = 0
x = -44 or x=42
Answer:
The consecutive even integers are either -44 and -42 or 42 and 44.
...
3. Two positive integers differ by 12 is best shown as: x and x+12
(x)(x+12)=405
x^2 + 12x -405 = 0
(x+27)(x-15) = 0
x=-27 or x=15
BUT we are told positive numbers only,
x = 15 and x+12 = 27
...
Done

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The Pythagorean theorem applies: the diagonal is the hypotenuse of a right triangle.
The sides are the triangle are the length and width of the rectangle.
We are told the diagonal is 10 m.
.


:: The length is 2 m > width, so the width + 2 = length
.

.
substitute
.

.

.

.
divide through by 2
.

.
factor
.

.
So our choices are W=-8 and W=6, but a negative width doesn't make sense.
.

.
checking our solution, is the diagonal 10?
.

.
So the
Correct
.
Answer:
The lengths of the sides of the rectangle are 6 and 8.
.
Done.

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Two consecutive numbers can be defined as x and y, but a better choice is x and x+1.
.
The product of the two numbers is: x(x+1). We are told this 41 more than the sum.
The sum of the two numbers is: x + (x+1).
.
So the equation we need to solve is:
.
x(x+1) = x+(x+1)+41
.
x^2 +x = 2x + 42
.
subtract 2x from both sides
.
x^2 +x -2x = 2x-2x + 42
x^2 -x = 42
.
subtract 42 from both sides
.
x^2 -x -42 = 42-42 = 0
.
factor
.
(x-7)(x+6) = 0
.
x = 7 or -6
.
We have to check both solutions by substituting these values into the equation.
.
7(7+1) = 7+(7+1)+41 ??
7*8 =56
7+8 + 41=56
Correct
.
-6(-6+1)=-6+(-6+1)+41 ??
36 -6 = -6 -6 + 42 ??
30 = 30
Correct.
.
Answer the consecutive integers are either 7 and 8 OR -6 and -5.
.
Done

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When you're dealing with fractions you need to get everything to the same denominator.
1/4 + 1/3 cannot be directly added using fractions.
The least common multiple of 4 and 3 is 12, so they can have the same denominator of 12.


So...

.
In this case, you're trying to add 9 3/4 + 8 5/6.
The least common multiple of 4 and 6 is 12. We can figure this out by multiplying 4*6=24, but we notice that 4=2*2 and 6=2*3, so 12 is the LEAST common multiple.
.
So...


.

.
But 19/12 is more than 1, it is

.

.
How do we know that 3/4 = 9/12?
.
Well, remember the idea is the 'least common multiple'.
.
To change 3/4 to 9/12 you simply look at how many times 4 goes into 12. 12/4 = 3
.
So you multiply 3/4 by 3/3 (Remember, you can multiply any number by 1, in this case 1 is 3/3.)
.

.
Similarly, how did 5/6 become 10/12?
It is the same process: 12/6 = 2
Multiply 5/6 by 1, which in this case is 2/2.
.

.
Done

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The slope-intercept form is: y = mx+b, where m is the slope and b is the y-intercept (the point were x=0).
.
The slope is defined as the rise over the run, which is the change in 'y' divided by the change in 'x'.
.
m = (y₂-y₁)/x₂-x₁)
.
(x₁,y₁) = (5,4)
(x₂,y₂) = (-4,-3)
.
m = (-3 -4)/(-4-5)
m = -7/-9 = 7/9
.
y = 7/9x + b
.
Substitute what we know already to find 'b'.
.
For example, we know the point (5,4) is on the line.
.
4 = 7/9(5) + b
.
Multiply through by 9 to remove the fraction
.
36 = 7(5) + 9b
36 = 35 + 9b
1 = 9b
9b = 1
b = 1/9
.
Now we can find the x-intercept, which is the point where y=0.
0 = 7/9x + 1/9
.
multiply through by 9
.
9(0) = 7x + 1
0 = 7x + 1
x = -1/7
(-1/7,0)
.
So, to summarize:
Equation is: y = 7/9x + 1/9
y-intercept is: (0, 1/9)
x-intercept is: (-1/7, 0)
.
Graphing it helps confirm your work.

.
Done