Questions: Solve the system of equations by graphing. 2x-y=2 x+2y=1

Transcript text: Solve the system of equations by graphing. \[ \begin{array}{l} 2 x-y=2 \\ x+2 y=1 \end{array} \]

Solution

Solution Steps

To solve the system of equations by graphing, we will first convert each equation into the slope-intercept form, \(y = mx + b\), and then graph them to find the point of intersection.

Step 1: Convert the First Equation to Slope-Intercept Form

The first equation is: \[ 2x - y = 2 \]

To convert it to slope-intercept form, solve for \(y\): \[ -y = -2x + 2 \] \[ y = 2x - 2 \]

Step 2: Convert the Second Equation to Slope-Intercept Form

The second equation is: \[ x + 2y = 1 \]

To convert it to slope-intercept form, solve for \(y\): \[ 2y = -x + 1 \] \[ y = -\frac{1}{2}x + \frac{1}{2} \]

Step 3: Graph the Equations

Now, graph the two equations:

\(y = 2x - 2\)
- This line has a slope of 2 and a y-intercept of -2.
\(y = -\frac{1}{2}x + \frac{1}{2}\)
- This line has a slope of \(-\frac{1}{2}\) and a y-intercept of \(\frac{1}{2}\).

Step 4: Find the Point of Intersection

To find the point of intersection, set the equations equal to each other: \[ 2x - 2 = -\frac{1}{2}x + \frac{1}{2} \]

Solve for \(x\): \[ 2x + \frac{1}{2}x = \frac{1}{2} + 2 \] \[ \frac{5}{2}x = \frac{5}{2} \] \[ x = 1 \]

Substitute \(x = 1\) back into one of the original equations to find \(y\). Using the first equation: \[ y = 2(1) - 2 = 0 \]

Thus, the point of intersection is \((1, 0)\).