Question

warm - up mentally find the slope of each linear equation.

1. y = 4 + 2x
2. 2y = 6 - 9x
3. 9x - 3y = 12
4. 1/3y+5x = 8

1 one, zero, infinitely many 5x - 2y = 10
here is an equation: 5x - 2y = 10.
create a second equation that would make a system of equations with
1 one solution 5x - 2y = 10
2 no solutions
3 infinitely many solutions

Explanation:

Step1: Recall linear - equation form

The general form of a linear equation is $y = mx + b$, where $m$ is the slope.

Step2: Rewrite given equations in slope - intercept form

For $y = 4+2x$

It is already in $y=mx + b$ form, and the slope $m = 2$.

For $2y=6 - 9x$

Divide both sides by 2: $y = 3-\frac{9}{2}x$. The slope $m=-\frac{9}{2}$.

For $9x - 3y=12$

First, isolate $y$: $-3y=-9x + 12$, then $y = 3x-4$. The slope $m = 3$.

For $\frac{1}{3}y+5x=\frac{2}{3}$

Isolate $y$: $\frac{1}{3}y=-5x+\frac{2}{3}$, then $y=-15x + 2$. The slope $m=-15$.

For creating second - equations for the system $5x - 2y=10$ (or $y=\frac{5}{2}x - 5$ with slope $m=\frac{5}{2}$):

One solution

A second equation with a different slope will give one solution. For example, $y=x + 1$ (or $x - y=-1$).

No solutions

A second equation with the same slope but a different y - intercept will give no solutions. For example, $5x-2y = 5$ (or $y=\frac{5}{2}x-\frac{5}{2}$).

Infinitely many solutions

The same equation (or a multiple of the original equation) will give infinitely many solutions. For example, $10x-4y = 20$ (which is $2(5x - 2y)=2\times10$).

Answer:

1. For finding slopes:

• $y = 4+2x$, slope is $2$.
• $2y=6 - 9x$, slope is $-\frac{9}{2}$.
• $9x - 3y=12$, slope is $3$.
• $\frac{1}{3}y+5x=\frac{2}{3}$, slope is $-15$.

1. For creating second - equations for the system $5x - 2y=10$:

• One solution: $x - y=-1$
• No solutions: $5x-2y = 5$
• Infinitely many solutions: $10x-4y = 20$