Questions: Graph the system below and write its solution. y=3x-2 x+2y=3

Graph the system below and write its solution. y=3x-2 x+2y=3
Transcript text: Graph the system below and write its solution. \[ \left\{\begin{array}{l} y=3 x-2 \\ x+2 y=3 \end{array}\right. \] Note that you can also answer "No solution" or "Infinitely many Explanation Check
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Solution

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Solution Steps

Step 1: Write the system of equations

The given system of equations is: \[ \begin{cases} y = 3x - 2 \\ x + 2y = 3 \end{cases} \]

Step 2: Graph the first equation

The first equation is \( y = 3x - 2 \). This is a linear equation in slope-intercept form \( y = mx + b \), where \( m = 3 \) (slope) and \( b = -2 \) (y-intercept).

  1. Plot the y-intercept \((0, -2)\) on the graph.
  2. Use the slope to find another point. From \((0, -2)\), move up 3 units and right 1 unit to get the point \((1, 1)\).
  3. Draw the line through these points.
Step 3: Graph the second equation

The second equation is \( x + 2y = 3 \). Rewrite it in slope-intercept form \( y = mx + b \).

  1. Solve for \( y \): \[ 2y = -x + 3 \] \[ y = -\frac{1}{2}x + \frac{3}{2} \]

  2. Plot the y-intercept \((0, \frac{3}{2})\) on the graph.

  3. Use the slope to find another point. From \((0, \frac{3}{2})\), move down 1 unit and right 2 units to get the point \((2, 0.5)\).

  4. Draw the line through these points.

Step 4: Find the intersection point

The solution to the system of equations is the point where the two lines intersect.

  1. Set the equations equal to each other to find the x-coordinate of the intersection: \[ 3x - 2 = -\frac{1}{2}x + \frac{3}{2} \]

  2. Solve for \( x \): \[ 3x + \frac{1}{2}x = \frac{3}{2} + 2 \] \[ \frac{7}{2}x = \frac{7}{2} \] \[ x = 1 \]

  3. Substitute \( x = 1 \) back into the first equation to find the y-coordinate: \[ y = 3(1) - 2 \] \[ y = 1 \]

Final Answer

The solution to the system of equations is the point \((1, 1)\).

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