Algebra

Super Easy

200

Simplify the expression 4m+5+2m-1

6m+4

Easy

200

What is the size of the smallest angle.

The smallest angle is 34 degrees.

The angles in a triangle add up to 180∘therefore we can write 

4𝑥+(2𝑥−10)+(3𝑥−8)=180

Now we have an equation we can solve.

9𝑥−18=180

9𝑥=198

𝑥=22∘

The angles are :

4×22=88∘

2×22−10=34∘

3×22−8=58∘

The smallest angle is therefore 34 degrees

300

Which of the following expressions has the smallest value when a=5 and b=-3?

(i) 1/2(a-b)

(ii) ab

(iii) b^2

(iv) b-4a

(iv) b-4a

Why? Because...

(i) 4

(ii) -15

(iii) 9

(iv) -23

400

The area of this triangle is 24𝑐𝑚 squared.

 

Find the perimeter of the triangle.

The area of a triangle is 𝑎𝑟𝑒𝑎=1/2×𝑏×ℎ 

If we fill in what we know we get:

24=1/2×6×(3𝑥−1)

24=3(3𝑥−1)

24=9𝑥−3

27=9𝑥

𝑥=3

Since 𝑥=3, the side lengths are 6𝑚, 8𝑐𝑚 and 10𝑐𝑚.

The perimeter is 6+8+10=24𝑐𝑚.

Medium

300

At a theme park the Jones family purchased 2 adult tickets and 3 child tickets for $48. The Evans family purchased 3 adult tickets and 1 child ticket for $44.

What is the cost of 1 child ticket?

We can write simultaneous equations to solve this.

2𝑎+3𝑐=48 (Equation 1)
3𝑎+𝑐=44 (Equation 2)

Multiply equation 2 by 3 to make the coefficients of c equal: 9𝑎+3𝑐=132 (Equation 3)

Subtract equation 1 from equation 3:
7𝑎=84
𝑎=12

Substitute a into equation 3:
3×12+𝑐=44
36+𝑐=44
𝑐=8

The cost of an adult ticket is $12 and a child ticket is $8.

Hard

200

x^3+4x^2−4x−16=0

What is x?

PS. ^ means to the power

x=2

300

{2x+3y=10

{4x−5y=20

x and y have the same values in both cases. Find x and y

x=5

y=0

500

Which of the following lines passes through the point (2, 5)?

(i) y=2x+5

(ii) y=4x-2

(iii) y=2x+1

(iv) y=2x-1

At the point (2, 5), x is 2 and y is 5. We can check which equation works when we substitute in these values:

y=2x+5

5=2×2+5

False

y=4x−2

5=4×2−2

False

y=2x+1

5=2×2+1

True

y=2x−1

5=2×2−1

False

So the correct answer is actually y=2x+1

Extreme

100

Find x+y, if: 5x+8y=67 and 2x-y=31

x+y=14 when x=15 and y=-1

Why? Because...

Given that system of the equations as :

5x + 8y = 67 and

2x - y = 31

Rearranging the terms in the above equation,

y = 2x - 31

Substitute the value of y in the equation 5x + 8y = 67,

5x + 8(2x - 31) = 67

5x + 16x - 248 = 67

21x = 67 + 248

21x = 315

x = 315/21

x = 15

Substitute the value of x in the equation 2x - y = 31,

2(15) - y = 31

30 - y = 31

y = -1

Substitute the values of x = 15 and y = -1 in the expression of x+y

⇒ x+y

⇒ 15+(-1)

⇒ 14

Hence, the value of expression of x+y is 14 when x = 15 and y = -1.

300

Ann and Kate have 80 dollars together. If Kate buys ice-cream for 5 dollars, then  Kate will have double Ann’s money. How much money does Ann have?

Ann has 25$. Why? Because...

1. Let the amount of money Ann has be A, and the amount Kate has be K.
2. It is given that A + K = 80 (equation 1).
3. The second condition states that after Kate buys ice-cream for 5 dollars, Kate's money will be double that of Ann's. So, if Kate currently has K dollars, after spending 5 dollars on ice-cream, she will have K - 5 dollars. This must be double the amount of Ann's money, or 2 times A. Thus, we get the equation K - 5 = 2A (equation 2).
4. Using substitution or elimination methods, we can solve these two equations simultaneously. If we express K from equation 1, we get K = 80 - A.
5. Now we substitute K in equation 2 with the expression from equation 1, getting 80 - A - 5 = 2A, which simplifies to 75 - A = 2A.
6. Adding A to both sides gives us 75 = 3A, and dividing both sides by 3, we get A = 25.