Question

section a: linear equations

1. solve:

\\(3x - 5 = 16\\)

1. solve:

\\(2(x + 4) = 18\\)

1. solve:

\\(5x + 7 = 2x + 19\\)

1. what is the slope between \\((2,3)\\) and \\((6,11)\\)?

1. write in slope-intercept form:

\\(y - 4 = 2(x - 3)\\)

Explanation:

Solve the first linear equation

We isolate the variable \(x\) by performing inverse operations.
First, add \(5\) to both sides:

$$3x = 16 + 5$$

$$3x = 21$$

Next, divide both sides by \(3\):

$$x = \frac{21}{3}$$

$$x = 7$$

Solve the second linear equation

We can solve this by first dividing both sides by \(2\):

$$x + 4 = \frac{18}{2}$$

$$x + 4 = 9$$

Subtract \(4\) from both sides to find \(x\):

$$x = 9 - 4$$

$$x = 5$$

Solve the equation with variables on both sides

First, subtract \(2x\) from both sides to group the variable terms:

$$5x - 2x + 7 = 19$$

$$3x + 7 = 19$$

Next, subtract \(7\) from both sides:

$$3x = 19 - 7$$

$$3x = 12$$

Finally, divide by \(3\):

$$x = \frac{12}{3}$$

$$x = 4$$

Calculate the slope between two points

The slope \(m\) between points \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (6, 11)\) is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substitute the given coordinates:

$$m = \frac{11 - 3}{6 - 2}$$

$$m = \frac{8}{4}$$

$$m = 2$$

Convert to slope-intercept form

Slope-intercept form is written as \(Y = mX + b\).
Distribute the \(2\) on the right side:

$$Y - 4 = 2x - 6$$

Add \(4\) to both sides to isolate \(Y\):

$$Y = 2x - 6 + 4$$

$$Y = 2x - 2$$

Answer:

No.	Problem	Answer
2	Solve: \(2(x + 4) = 18\)	\(x = 5\)
3	Solve: \(5x + 7 = 2x + 19\)	\(x = 4\)
4	What is the slope between \((2,3)\) and \((6,11)\)?	\(2\)
5	Write in slope-intercept form: \(Y - 4 = 2(x - 3)\)	\(Y = 2x - 2\)