Question

question 1
solve the following equation for the value of the variable:
3(x + 2) - 4x = -3(2x + 5) + 1

question 2
find the slope of the line:
-6x - 3y = -12

question 3
find the slope of the line through the pair of points:
(-1, -1) and (7, 5)

Explanation:

Question 1

Step 1: Expand both sides

First, expand the expressions on both sides of the equation \(3(x + 2)-4x=-3(2x + 5)+1\).
Using the distributive property \(a(b + c)=ab+ac\), we get:
Left side: \(3x+6 - 4x\)
Right side: \(-6x-15 + 1\)
So the equation becomes \(3x + 6-4x=-6x-15 + 1\).

Step 2: Simplify both sides

Simplify the left side: \(3x-4x+6=-x + 6\)
Simplify the right side: \(-6x-14\)
Now the equation is \(-x + 6=-6x-14\).

Step 3: Add \(6x\) to both sides

Add \(6x\) to both sides to get all \(x\) terms on one side:
\(-x+6x + 6=-6x+6x-14\)
Which simplifies to \(5x + 6=-14\).

Step 4: Subtract 6 from both sides

Subtract 6 from both sides:
\(5x+6 - 6=-14 - 6\)
\(5x=-20\).

Step 5: Divide by 5

Divide both sides by 5:
\(x=\frac{-20}{5}=-4\).

Step 1: Rewrite in slope - intercept form

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
We start with the equation \(-6x-3y=-12\).
First, add \(6x\) to both sides: \(-3y=6x-12\).
Then, divide every term by \(-3\): \(y=\frac{6x}{-3}-\frac{12}{-3}\).

Step 2: Simplify the equation

Simplify the right - hand side: \(y=-2x + 4\).
Comparing with \(y = mx + b\), we see that \(m=-2\).

Step 1: Recall the slope formula

The slope \(m\) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step 2: Identify the points

Let \((x_1,y_1)=(-1,-1)\) and \((x_2,y_2)=(7,5)\).

Step 3: Substitute into the formula

Substitute \(x_1=-1\), \(y_1=-1\), \(x_2 = 7\), and \(y_2=5\) into the slope formula:
\(m=\frac{5-(-1)}{7-(-1)}=\frac{5 + 1}{7 + 1}=\frac{6}{8}=\frac{3}{4}\).

Answer:

\(x = - 4\)

Question 2