Question

question 1
find the slope of the following:
-2y + 3x = 4

question 2
find the slope of the line through each pair of points:
(2, 5) and (3, -4)

question 3
solve the following equation for the value of the variable.
\frac{1}{2}(2x - 4) - (x + 2) + 3x = 2(x - 5)

Explanation:

Question 1

Step1: Rewrite in slope - intercept form ($y = mx + b$)

Start with the equation $-2y+3x = 4$. First, isolate the $y$ - term. Subtract $3x$ from both sides: $-2y=-3x + 4$.

Step2: Solve for $y$

Divide every term by $-2$: $y=\frac{-3x}{-2}+\frac{4}{-2}$, which simplifies to $y=\frac{3}{2}x - 2$. In the slope - intercept form $y = mx + b$, the slope $m$ is $\frac{3}{2}$.

Step1: Recall the slope formula

The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Identify the points

Let $(x_1,y_1)=(2,5)$ and $(x_2,y_2)=(3,-4)$. Substitute these values into the slope formula: $m=\frac{-4 - 5}{3 - 2}$.

Step3: Calculate the slope

Simplify the numerator and the denominator: $m=\frac{-9}{1}=-9$.

Step1: Simplify the left - hand side (LHS) and right - hand side (RHS)

• Simplify LHS:
• First, distribute $\frac{1}{2}$ in $\frac{1}{2}(2x - 4)$: $\frac{1}{2}(2x)-\frac{1}{2}(4)-x - 2+3x$.
• $\frac{1}{2}(2x)=x$, $\frac{1}{2}(4) = 2$. So LHS becomes $x-2 - x - 2+3x$.
• Combine like terms: $(x - x+3x)+(-2 - 2)=3x-4$.
• Simplify RHS:
• Distribute 2 in $2(x - 5)$: $2x-10$.

Step2: Solve for $x$

Now we have the equation $3x-4 = 2x-10$. Subtract $2x$ from both sides: $3x-2x-4=2x - 2x-10$, which gives $x - 4=-10$.

• Then add 4 to both sides: $x-4 + 4=-10 + 4$.

Step3: Find the value of $x$

Simplifying, we get $x=-6$.

Answer:

$\frac{3}{2}$

Question 2