Question

number of solutions. (examples 1-3)

1. 3y - 4 = 6x y = 2x - 3

m = ____ m = ____
b = ____ b = ____

circle: one, none, infinite

Explanation:

Step1: Rewrite first equation in slope - intercept form

The slope - intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.
For the equation $3y-4 = 6x$, we solve for $y$:
Add 4 to both sides: $3y=6x + 4$
Divide both sides by 3: $y=\frac{6x + 4}{3}=2x+\frac{4}{3}$
So for the equation $3y - 4=6x$ (or $y = 2x+\frac{4}{3}$), the slope $m = 2$ and the y - intercept $b=\frac{4}{3}$.

Step2: Identify slope and y - intercept of the second equation

The second equation is $y = 2x-3$. Comparing it with $y=mx + b$, we can see that the slope $m = 2$ and the y - intercept $b=- 3$.

Step3: Determine the number of solutions

Two linear equations $y = m_1x + b_1$ and $y=m_2x + b_2$ have:

• One solution if $m_1

eq m_2$

• No solution if $m_1 = m_2$ and $b_1

eq b_2$

• Infinite solutions if $m_1=m_2$ and $b_1 = b_2$

Here, $m_1 = 2$, $m_2 = 2$ (so slopes are equal) and $b_1=\frac{4}{3}$, $b_2=-3$ (y - intercepts are not equal). So the system of equations has no solution.

Answer:

For $3y - 4 = 6x$ (or $y = 2x+\frac{4}{3}$): $m=\boldsymbol{2}$, $b=\boldsymbol{\frac{4}{3}}$
For $y = 2x-3$: $m=\boldsymbol{2}$, $b=\boldsymbol{-3}$
Number of solutions: \boxed{none}