Graph the system below and write its solution. $$ \begin{array}{l}y=-\frac{1}{2} x+3 \ -x-2 y=-2\end{array} $$

Question

UpStudy AI · Accepted Answer

Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}y=-\frac{1}{2}x+3\-x-2y=-2\end{array}\right.$$
- step1: Substitute the value of $$y:$$
  $$-x-2\left(-\frac{1}{2}x+3\right)=-2$$
- step2: Simplify:
  $$-6=-2$$
- step3: Calculate:
  $$\left\{ \begin{array}{l}x \in \varnothing \y \in \varnothing \end{array}\right.$$
- step4: Rewrite:
  $$(x, y) \in \varnothing$$
Let's analyze the system of equations given:

1. The first equation is:
   $$
   y = -\frac{1}{2} x + 3
   $$
   This is a linear equation in slope-intercept form, where the slope $$ m = -\frac{1}{2} $$ and the y-intercept $$ b = 3 $$.

2. The second equation is:
   $$
   -x - 2y = -2
   $$
   We can rearrange this into slope-intercept form by solving for $$ y $$:
   $$
   -2y = x - 2 \implies y = -\frac{1}{2} x + 1
   $$

Now we have the two equations:
1. $$ y = -\frac{1}{2} x + 3 $$
2. $$ y = -\frac{1}{2} x + 1 $$

### Step 1: Graphing the Equations
Both equations have the same slope of $$ -\frac{1}{2} $$, which means they are parallel lines. The first line intersects the y-axis at $$ (0, 3) $$ and the second line intersects at $$ (0, 1) $$.

### Step 2: Finding the Solution
Since the two lines are parallel, they will never intersect. Therefore, there is no solution to this system of equations.

### Conclusion
The solution to the system of equations is:
$$
\text{No solution}
$$

If you need a visual representation, please use graphing software or tools to plot the lines based on the equations provided.
There is no solution to the system of equations because the two lines are parallel and do not intersect.

Graph the system below and write its solution. \[ \begin{array}{l}y=-\frac{1}{2} x+3 \\ -x-2 y=-2\end{array} \]

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions