Any Mathematical Sentence That States That Two Quantities Are Equal Is Called an Equation

MPM1D Grade 9

Solving Equations

Any mathematical sentence that states that two quantities are equal is called an equation.

Example: 5x + 6 = 2(x + 5) + 5 is an equation. The expressions 5x + 6 is on the left side and 2(x + 5) + 5 is on the right side.

Example: x = 3 is the solution to the equation 5x + 6 = 2(x + 5) + 5. When x is replaced with 3, both sides of the equation result in 21.

L.S. = 5(3) + 6 R.S. = 2(3 + 5) + 5

= 15 + 6 = 2(8) + 6

= 21 = 16 + 5

= 21

A linear equation in one variable is solved by isolating the variable. Before this can be done,

· All brackets must be eliminated using the distributive property

a(b + c) = ab + ac

· All fractions must be eliminated by multiplying each term of the equation by the lowest common denominator

Example 1 Solve: 13x + 9 = 11x + 5.

Solution

13x + 9 = 11x + 5

13x – 11x = 5 – 9

2x = –4

x = –2

Example 2: Solve: x + 6(x – 3) = 2(3x – 2)

Solution

x + 6(x – 3) = 2(3x – 2)

x + 6x – 18 = 6x – 4

x + 6x – 6x = 18 – 4

x = 14

Example 2: Solve: 34 t – 2 = 12 (t + 2)

Solution

34 t – 2 = 12 (t + 2)

4 × 34 t – 4 × 2 = 4 × 12 (t + 2)

3t – 8 = 2(t + 2)

3t – 8 = 2t + 4

3t – 2t = 8 + 4

t = 12

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PRACTICE

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1. Solve.

(a) 3y+5=11 (b) 4x-3= -11

(e) 9p-10=6+p (f) 2m+6.1=16.5

(g) 4a-2.8=6.8 (h) 15.8-6m=3.8

(i) 8y-6.9=3y+3.6 (j) 12.8-3m=8m-33.4

2. Find the root of each equation.

(a) 3n+4=5n

(b) 3x-10=2x-3

(d) 4c-2=3c+1

(e) 8m-1=4m+4

(f) 43-r=52r+1

(g) 122m-3=2m+4

(h) 0.5x+2=0.1x+0.6x-3

(i) 6.5x-3=2.43-x

3. Solve.

(a) x2 = 4 (b) 3x5= -9

(e) 14x-3=4 (f) 45x-3=5

(g) 7=1+ 23x (h) -5+ 14x= -7

4. Solve.

(a) 2y+ 12= 23

(b) 76x-2= 13

(d) 3- m2=5- m3

(e) 23y-3= 45y-5

(f) 12x+ 13x=10

(g) 34x- 18x=5

5. Solve.

(a) 132x-1=1 (b) 162y+6=2

(e) 12x-1= 14x+1

(f) 4x - 15= 2x + 32

(g) y - 73= y - 24

6. Determine64, the value of the unknown variable using the given formula and information.

a) A = lw, A = 464, w = 4, l = ?

b) A = 12bh, A = 288, h = 24, b = ?

c) A = 12 (a + b) × h, A = 96, a = 4, b = 8, h = ?

d) P = 2l + 2w, P = 80, l = 8, w = ?

e) A = s2, A = 625, s = ?

f) I = 4.5A + 2.5S, A = 120, I = 700, S =?

7. At the December concert, 209 tickets were sold. There were 23 more student tickets sold than twice the number of adult tickets. How many of each were sold?

8. A rectangle with a perimeter of 54 cm is 3 m longer than it is wide. What are its length and width?