###### tags: `MATA02-2022`
# Ungraded problems 1

Below we remind you of some high school mathematical skills that will be necessary in this course. At the bottom are some practice problems for solving linear equations. You will not turn these in, and they will not be marked. However, you should ensure that you can do them.

## A) Order of operations
In the order of operations, by convention, multiplication and division are completed before addition and subtraction.

**Example**:
- Correct: $3 \times 4 - 10 \div 2 = (3 \times 4) - (10 \div 2) = 12 - 5 = 7$
- Incorrect: $3 \times 4 - 10 \div 2 = ((3 \times 4) - 10) \div 2) = (12 - 10) \div 5 = 2 \div 2 = 1$
- Incorrect: $3 \times 4 - 10 \div 2 = 3 \times (4 - 10) \div 2=3 \times (-6) \div 2 = -18 \div 2 = -9$

## B) Distributive property
Distributive property: If $a, b, c, d$ are real numbers, then:
$$
a(b+c) = a \times (b+c) = (a\times b)+(a\times c) = ab + ac
$$
Similarly,
$$
a(b+c+d) = a \times (b+c+d) = (a\times b)+ (a\times c) + (a\times d) = ab + ac + ad
$$
and so on.

## C) Factoring or distributive property in reverse

If $a, b, c, d$ are real numbers, then:
$$
ab + ac = (a\times b)+(a\times c) = a \times (b+c) = a(b+c)
$$
or
$$
ac + bc = (a\times c)+(b\times c) = (a+b) \times c = (a+b)c
$$

**Example 1**: $39 + 18 = 3 \times 13 + 3 \times 6 = 3 \times (13+6)$

**Example 2**: $32a + 24ab - 16ac = 8a \times 4 + 8a \times 3b - 8a \times 2c = 8a(4 + 3b - 2c)$

## C) Solving for an unknown in linear equations
In order to solve for an unknown variable, say $x$, you must isolate that variable on one side of the equation. Generally, this will be done by doing the same operation to both sides of the equation and transforming using various mathematical properties such as the ones above until you are left with the variable of interest as the only thing on one side.

**Example 1**: Solve for $x$ in the following equation: $x-6=10$.
- Add $6$ to both sides of the equation $x-6 + 6 = 10 + 6$ to get $x = 16$.

**Example 2**: Solve $4x - 6 = 10$
- Add $6$ to both sides of the equation: $4x = 16$
- Divide both sides by $4$: $x = 4$

**Example 3**: Solve $4x - 6 = 12x - 30$
- Subtract $4x$ from both sides: $-6 = 8x - 30$
- Add $30$ to both sides: $24 = 8x$
- Divide by $8$ on both sides: $3 = x$

**Example 4**: Solve for $x$: $\frac{3}{4}x + \frac{5}{6} = 5x - \frac{125}{3}$
- Multiply both sides by $6$: $6\left(\frac{3}{4}x + \frac{5}{6}\right) = 6\left(5x - \frac{125}{3}\right)$
- Which simplifies to: $\frac{18}{4}x + 5 = 30x - \frac{6 \cdot 125}{3}$.
- Note: $\frac{18}{4} = \frac{9\cdot 2}{2 \cdot 2} = \frac{9}{2}$ and $\frac{6 \cdot 125}{3} = \frac{2 \cdot 3 \cdot 125}{3} = 2 \cdot 125 = 250$
- Use those simplifications to get: $\frac{9}{2}x + 5 = 30x - 250$
- Multiply both sides by 2: $2\left(\frac{9}{2}x+5 \right)= 2(30x - 250)$
- Simplify: $9x + 10 = 60x - 500$.
- Subtract $9x$ from both sides: $10 = 51x - 500$
- Add 500 to both sides: $510 = 51x$
- Divide both sides by 51: $x = \frac{510}{51} = 10$
- The answer is $x = 10$

## Practice solving linear equations

Solve for $x$ in the following:
1. $x + 7 = 19$
2. $3(x+5) = 24$
3. $3(2x+4) = 54$
4. $2(5+x-3x)=6$
5. $4x - 7 = 12x - 31$
6. $5x - 17 = 3x + 2$
7. $\frac{2}{3}x + \frac{5}{7} = \frac{1}{2}x + \frac{22}{21}$
8. $\frac{3}{7}x + \frac{2}{35} = \frac{5}{3}x - \frac{14}{15}$

<details>
<summary>Solutions (click to expand)</summary>

1. $x = 12$
2. $x = 3$
3. $x = 7$
4. $x = 1$
5. $x = 3$
6. $x = \frac{19}{2}$
7. $x = 2$
8. $x = \frac{4}{5}$

</details>