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First, get the main equation into slope-intercept form: y = mx + b
.
3x -y = 8
-y = -3x +8
y = 3x - 8
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Second, you recall that any line with slope = 3 will be parallel to this line.
Your problem is define such a line that goes through the point (7,2).
Well, plug x=7 into the equation and then adjust 'b' to make y=2.
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y = 3x -b
2 = 3(7) -b
2 = 21 -b
-19 = -b
b = 19
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so,
y = 3x -19 will be parallel and will go through the point.
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Check by graphing the two equations. y=3x-8 is red; y=3x-19 is green.

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Rechecking the question, this equation needs to be shown in slope-intercept and in standard form.
y = 3x-19 :: slope-intercept form
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Standard form is: Ax + By = C
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subtract 3x from both sides
-3x + y = -19
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multiply both sides by -1
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3x -y = 19 :: standard form
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Done

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Rearrange the equation into: y = whatever format
.

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subtract x^2 from both sides
.

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divide both sides by 2
.

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So you want to know if (2,-3) is on the graph? Plug the value of 'x' into the equation and see.
.




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Yes, the point (2,-3) is on the graph.
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Done

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Substitution is done by...well...substitution.
You use one equation to find a value for 'x'.
Then you substitute that into the other equation.
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5x + 8y = 18
x = 6 -2y
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So the second equation is a easy substitution candidate into the first equation.
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5(6-2y) +8y = 18
30 -10y +8y = 18
30 -2y = 18
-2y = -12
y = 6
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Now substitute back to find 'x'.
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x = 6 -2(y)
x = 6 -2(6)
x = 6 -12
x = -6
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Check by plugging these values into the other equation to ensure they're right.
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5x + 8y = 18
5(-6) + 8(6) = 18??
-30 +48 = 18
Correct!
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Answer:
x = -6
y = 6
.
Done.

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Looks like a perimeter, where B = base of the figure and H = height.

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'x' cannot = 0 because you cannot divide by 0.

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Three consecutive numbers could be called: a, b, c.
But it is far better to define them as:



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:: the product of the two smaller is 40 less than the square of the largest
.







.
Checking...


So

Correct.
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Answer:
The three consecutive numbers are: 12, 13, and 14.
.
Done.

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Perimeter = P = 2L + 2W
P = 52
L = W+12 :: Length is 12 more than width, so you'd have to add 12 to L to make it equal.
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2L + 2W = 2(L+W)
2(L+W) = 52
L+W = 26
(W+12)+W = 26
2W+12=26
2W = 14
W = 7
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L=W+12
L=7+12
L=19
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Check perimeter to be sure it is 52.
2(7+19) = 2(26) = 52
.
State answer:
W = 7
L = 19
.
Done

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3x - 2y = 10
5x + 3y = 4
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It is presents itself as a system of linear equations.
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Multiply the first equation by 3.
Multiply the second equation by 2.
Add the equations to eliminate the 'y' term.
Solve for 'x'.
Substitute to find 'y'.
Check.
Done.

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x = number of apples in the bag originally
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We are told the first man ate 1/3 of the apples.
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That means that 2/3 of the apples were left in the bag.
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We are told the second man ate 1/3 of the apples that he found in the bag.
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BUT notice that he did NOT eat 1/3 of x apples.
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He ate 1/3 of the 2/3 of the apples that were left in the bag.
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1/3*2/3 = 2/9
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So the second man ate 2/9 of the x total apples.
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That means the two men ate a total of 1/3x plus 2/9x, leaving 8 apples in the bag.
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1/3x + 2/9x = 3/9x + 2/9x = 5/9x
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However, notice that the above tells us how many apples they ate, not how many apples are left.
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x - 5/9x is how many apples are left, which is 9/9x - 5/9x = 4/9x.
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4/9x is how many apples are left.
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4/9x = 8
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4x = 72
x = 18
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Checking our work...
How many apples are 1/3 of 18?
1/3*18 = 6
So the first man ate 6 apples, leaving 12 in the bag.
How many apples are 1/3 of 12?
1/3*12 = 4
So the second man ate 4 apples, leaving 8 apples in the bag.
Correct.
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Answer:
There were 18 apples in the bag originally.
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Done.

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x = one number
y = another number
x + y = 45 :: given
x = y + 9 :: y is 9 less than x
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substitute x = y+9 in the first equation
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y+9 + y = 45
2y = 36
y = 18
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now substitute y=18 in the second equation
x + y = 45
x + 18 = 45
x = 27
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checking...
Is the smaller number 9 less than the larger number?
Yes.
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Answer:
The two numbers are 18 and 27.
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Done

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Start by getting the equation into slope-intercept form: y = mx + b.
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5x -8y = -15
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subtract 5x from both sides...
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-8y = -5x -15
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divide both sides by -8
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y = -5/(-8)x - 5/(-8)
y = 5/8x + 5/8
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The y-intercept is the point where x=0.
y = 5/8(0) + 5/8
y = 5/8
(0, 5/8)
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The x-intercept is the point where y=0.
0 = 5/8x + 5/8
x = -1
(-1,0)
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Graphing is a good way to check.

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The slope of 3 means that for a change of 1 'x' there will be a change of 3 'y'.
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m = rise/run = 3/1
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We have the point (1,1), so the next point would be (2,4). With two points, you can graph a straight line.
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Or we can address the problem in terms of y = mx + b, which is the slope-intercept form of an equation.
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y = 3x + b
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But then the question is, "What is the value 'b' to be sure it goes through the point (1,1)?"
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Well, (1,1) means that when x=1, y =1, so we can show this as:
1 = 3(1) + b
b = -2
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This suggests the equation of the line with slope = 3 that goes through the point (1,1) is:
y = 3x -2
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Graphing it we can see...

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That looks correct.
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We also can do additional checking.
Some other points of interest are the y-intercept and the x-intercept.
The y-intercept is where x=0.
y = 3(0)-2 = -2
(0, -2)
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The x-intercept is where y=0.
0 = 3x -2
x = 2/3
(2/3, 0)
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By inspecting the graph, you can see these points are on the line, too.
.
Done

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The slope, 'm', is defined as the rise divided by the run.
The rise is the change in 'y'.
The run is the change in 'x'.
The change in 'y' is defined as y₂ - y₁.
The change in 'x' is defined as x₂ - x₁.
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We have two points:
(-3, 5) = (x₁, y₁)
(6, -1) = (x₂, y₂)
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y₂ - y₁ = -1 -5 = -6
x₂ - x₁ = 6 - (-3) = 9
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m = -6/9 = -2/3
.
Done

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The general form of a quadratic equation that has been solved is to have two factors that are multiplied. In this case, you're told the solutions are x= -3.5 and x=2.2. So the factored form of the equation would be:
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(x + 3.5)(x - 2.2) = 0
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Using FOIL...
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x^2 -2.2x +3.5x -7.7 = 0
x^2 + 1.3x - 7.7 = 0
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multiply by 10 to get rid of the decimals...
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10x^2 + 13x - 77 = 0
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Checking this equation using the quadratic formula, you can determine if the solutions are x= -3.5 and x= 2.2...
.
.

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=3249 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 2.2, -3.5. Here's your graph:

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Done

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b = number of balcony tickets sold
o = number of orchestra tickets sold
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b + o = 195 :: given
or,
b = 195 - o and o = 195-b
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125o = value of the orchestra tickets sold
70b = value of the balcony tickets sold
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125o + 70b = 26650 :: given
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substitute b = 195 - o:
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125o + 70(195 - o) = 26650
125o + 13650 - 70o = 26650
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collect and simplify
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125o - 70o = 26650 - 13650
55o = 13000
o = 236.36
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That is an odd result because it calls for selling a fraction of a ticket, and it calls for selling more than 195 tickets.
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If ONLY orchestra tickets were sold @ $125 each, then that would be 213.2 tickets, which is more than 195.
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So there appears to be something wrong with your problem statement.
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Done

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The given equations are...
3x -2y = 3
-x + y = 1
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Multiply the second equation by 2 to set up for elimination...
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-2x + 2y = 2
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Then add the two equations...
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3x -2y = 3
-2x +2y = 2
x = 5
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Substitute x=5 into first equation to find y...
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3x - 2y = 3
3(5) -2y = 3
15 -2y = 3
-2y = 3 -15
y = -12/-2 = 6
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Check answer by substituting into the 2nd equation...
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-x + y = 1
-5 + 6 = 1
Correct!
.
Answer:
x = 5
y = 6
.
Done

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The area of a triangle is 1/2*base*height.
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b = base = (x+1) ft
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h = height = 8x ft
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Area = 1/2*(x+1)*8x ft^2
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A = 1/2(8x^2+8x) ft^2
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A = 4x^2 + 4x ft^2
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A = 4x(x+4) ft^2
.
Done

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Perimeter of a rectangle = 2*length + 2*width.
l = length
w = width
.
P = 132 :: given
.
l = w+46 :: given
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Substituting into the perimeter equation...
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2l + 2w = 132
2(w+46) + 2w = 132
2w + 92 + 2w = 132
4w + 92 = 132
4w = 40
w = 10
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l = w+46 = 56
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Checking this answer...
.
P = 2*56 + 2*10
P = 112 + 20
P = 132
Correct.
.
Answer:
Length = 56
Width = 10
.
Done

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When doing sum of digits problems, you have to remember that numbers like 'xy' are not being multiplied.
Consider the number 23.
It really is not "one" number, but instead it means 2*10 + 3*1.
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So the number is 'xy' but that is NOT x*y, it's just 'x' next to 'y'.
The value of the number 'xy' = 10x + y.
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x + y = 14 :: given
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36 is a value, so to subtract it, you have to calculate the value of xy:
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10x+y -36 = 10y + x
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Setting up a system of equations:
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10x + y - 36 - 10y - x = 0
9x - 9y = 36
x - y = 4
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x + y = 14
x - y = 4
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adding
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2x = 18
x = 9
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x + y = 14
given x = 9
y = 5
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so the original number, 'xy', appears to be 95.
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checking...
'yx' would be 59.
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so, is 95 - 36 = 59?
Yes, it is.
.
Answer:
The number is 95.
.
Done.

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First, get the equation into slope-intercept form: y = mx+b
.
3x + 5y = 15
5y = -3x + 15
y = -3/5x+ 3
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The y-intercept is where x=0, which equals 3. The point is (0, 3).
.
The x-intercept is where y = 0.
0 = -3/5x + 3
x = 5
The point is: (5,0)
.
With two points, you can draw a line.
.

.
Done

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37% = .37
.
x = some number
.
So, we can state algebraically:
.37x

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x + y = 7
x = 7-y
.
x^2 + y^2 = 29
.
(7-y)^2 + y^2 = 29
(7-y)(7-y) +y^2 = 29
49 -7y - 7y + y^2 + y^2 = 29
2y^2 -14y + 49 = 29
2y^2 - 14y + 49 - 29 = 0
2y^2 - 14y + 20 = 0
y^2 - 7y + 10 = 0
(y-2)(y-5) = 0
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y=2 OR y=5
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But it is possible that one of these is NOT an answer.
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Checking...
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x+y = 7
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Substituting y = 2, then x = 5
2^2 = 4
5^2 = 25
4+25 = 29
So, y=2 and x=5 is a solution.
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Substituting y=5, then x = 2.
That works too.
.
Answer:
The two numbers are 2 and 5.
.
Done

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.


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i is an imaginary number that represents
.

.



.
Done

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We need more information to solve this.
Is there a total number of minutes of commercials per hour?
Is there some ratio of 30-sec and 60-sec commercials?
As stated, the DJ could have played 14 30-sec commericals or 14 60-sec commericals, or any combination.

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d = r*t is the basic distance equation.
s = speed of the motor boat in calm water
c = speed of current
s + c = speed of the boat traveling with the current
s - c = speed of the boat traveling against the current
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traveling with the current the boat went 36 miles in 2 hrs
d = 36
r = s+c
t = 2
36 = (s+c)*2
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traveling against the current the boat can return 36 miles in 3 hrs
d = 36
r = s-c
t = 3
36 = (s-c)*3
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We've got two unknowns and we've got two unknowns, so this can be approached as a system of linear equations.
.
2s + 2c = 36
3s - 3c = 36
.
Multiply the first equation by 3 and the second by 2 to enable elimination.
.
3(2s + 2c) = 3(36) = 6s + 6c = 108
2(2s - 2c) = 2(36) = 6s - 6c = 72
.
6s + 6c = 108
6s - 6c = 72
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adding them to eliminate the 6c and -6c.
.
12s = 180
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divide by 12
.
s = 15 = speed of the boat in calm water
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substituting back into our first equation
.
36 = (s+c)*2
2s + 2c = 36
2(15) + 2c = 36
2c = 6
c = 3 = speed of the current
.
checking...
.
traveling with the current, the boat's speed = 15+3 = 18
to travel 36 miles, it would take 2 hr, which is what the problem stated.
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traveling against the current, the boat's speed = 15 -3 = 12
again we can see that to travel the 36 miles it would take 3 hrs, which is also what was stated.
.
Answer:
s = 15 = speed of the boat in calm water
c = 3 = speed of the current
.
Done

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The area of the patio is x*x = x^2.
The area of the table is 3*6 = 18.
The area of the patio remaining is x^2 - 18.

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You cannot solve one equation with two unknowns.
With two equations with two unknowns, you can set up a system of equations and add or subtract them. That is the elimination step.

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Four consecutive positive integers can be described as a, b, c, and d.
But it is better to define them in terms of one another.
a = x
b = x+1
c = x+2
d = x+3
.
We are told the sum is 5 times the smallest number.
.
x + (x+1) + (x+2) + (x+3) = 5x
.
collecting terms...
4x + 6 = 5x
6 = x
x = 6
.
So the possible consecutive positive integers are 6, 7, 8 and 9.
.
Checking the solution, is the sum = 5*6 = 30?
6+7+8+9 =30
Correct!
.
Answer:
The consecutive positive digits are 6, 7, 8 and 9.
.
Done

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w = width
l = length
w = 1/2*l - 6
P = 2(l+w) = 72
2l + 2w = 72
.
substitute w = 1/2*l -6
.
2l + 2w= 72
2l + 2(1/2*l - 6) = 72
2l + l - 12 = 72
3l = 84
l= 28
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w = 1/2*l -6 = 1/2*28 - 6 = 14-6 = 8
.
checking
2l + 2w = 72?
2l = 2*28 = 56
2w = 2*8 = 16
56+16= 72
Correct.
.
Answer:
length = 28
width = 8
.
Done

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A key part of solving these types of problems is keeping track of how much 'pure' stuff you need.
You need 1080 mL of 50% solution = .5*1080 = 540 mL of pure acid.
.
x = volume of 70% acid you will use
y = volume of 30% acid you will use
.
BUT since you know x+y = 1080, it is better to define the volumes in terms of 'x' only
.
y = 1080-x
.
Set up an equation that will describe the two parts and the final result
.
.7x + .3(1080-x) = .5*1080
.
multiply by 10 to get rid of decimals
7x + 3(1080-x) = 5*1080 = 5400
.
expand, collect, simplify...
.
7x + 3240 - 3x = 5400
4x = 2160
x = 540
.
y = 1080 - 540 = 540
.
So our proposed solution is 540 mL of 70% acid and 540 mL of 30% solution.
By inspection that makes sense because (.7+.3)/2 = .5
But we need to check our work...always...
.
How much 'pure' acid will be in the final solution compared to how much you need.
You need 540 mL.
.
.7*540 = 378
.3*540 = 162
378 + 162 = 540
Correct!
.
Answer:
You need to mix 540 mL of 70% acid with 540 mL of 30% acid to produce 1080 mL of 50% acid.
.
Done

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x = one number
y = another number
x = y+5 :: x is 5 greater than y
x*y = 36
.
substitute x = y+5
.
(y+5) * y = 36
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expand, collect, simplify, factor, solve...
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good luck!