Graph using Des mos, then find the solution. y=(2)/(3)x+(1)/(3)y=x-1

To solve the system of equations using the Desmos graphing tool, follow these steps:<br /><br />1. **Open Desmos**: Go to [Desmos Graphing Calculator](https://www.desmos.com/calculator).<br /><br />2. **Enter the first equation**: In the first input box, type `y = (2/3)x + (1/3)y` and press Enter.<br /><br />3. **Simplify the first equation**: To make it easier to graph, simplify the first equation. Combine like terms:<br /> \[<br /> y = \frac{2}{3}x + \frac{1}{3}y \implies \frac{2}{3}x + \frac{1}{3}y - y = 0 \implies \frac{2}{3}x - \frac{2}{3}y = 0 \implies 2x - 2y = 0 \implies x = y<br /> \]<br /> So, the first equation simplifies to \(y = x\).<br /><br />4. **Enter the second equation**: In the second input box, type `y = x - 1` and press Enter.<br /><br />5. **Find the intersection**: The solution to the system of equations is the point where the two lines intersect.<br /><br />6. **Verify the solution**: From the graph, you should see that the lines intersect at the point \((1, -1)\). This is the solution to the system of equations.<br /><br />To find the solution algebraically, solve the system of equations:<br />\[<br />\begin{cases}<br />y = x \\<br />y = \frac{2}{3}x + \frac{1}{3}y - 1<br />\end{cases}<br />\]<br /><br />Substitute \(y = x\) into the second equation:<br />\[<br />x = \frac{2}{3}x + \frac{1}{3}x - 1<br />\]<br />Combine like terms:<br />\[<br />x = \frac{2}{3}x + \frac{1}{3}x - 1 \implies x = \frac{3}{3}x - 1 \implies x = x - 1<br />\]<br />This simplifies to:<br />\[<br />0 = -1<br />\]<br />This is a contradiction, indicating that there is no solution to the system of equations. Therefore, the lines are parallel and do not intersect.<br /><br />In conclusion, the system of equations has no solution because the lines are parallel.