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:

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<reasoning>

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 \(Y\):
\[Y = 2x - 6 + 4\]
\[Y = 2x - 2\]
</reasoning>

<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: \(Y - 4 = 2(x - 3)\)	\(Y = 2x - 2\)

</answer>

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Answer:

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<reasoning>

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 \(Y\):
\[Y = 2x - 6 + 4\]
\[Y = 2x - 2\]
</reasoning>

<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: \(Y - 4 = 2(x - 3)\)	\(Y = 2x - 2\)

</answer>

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