To solve the given system of equations by graphing, we need to rewrite each equation in slope-intercept form, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept. Let's start with the given equations:
1. [tex]\(2x - y = 9\)[/tex]
2. [tex]\(6x - 3y = 9\)[/tex]
Step 1: Convert each equation to slope-intercept form.
Equation 1: [tex]\(2x - y = 9\)[/tex]
Isolate [tex]\(y\)[/tex]:
[tex]\[ 2x - y = 9 \][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ -y = -2x + 9 \][/tex]
Multiply everything by [tex]\(-1\)[/tex] to solve for [tex]\(y\)[/tex]:
[tex]\[ y = 2x - 9 \][/tex]
Equation 2: [tex]\(6x - 3y = 9\)[/tex]
Isolate [tex]\(y\)[/tex]:
[tex]\[ 6x - 3y = 9 \][/tex]
Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[ -3y = -6x + 9 \][/tex]
Divide everything by [tex]\(-3\)[/tex] to solve for [tex]\(y\)[/tex]:
[tex]\[ y = 2x - 3 \][/tex]
Step 2: Plot both equations on a graph.
- For the first equation [tex]\( y = 2x - 9 \)[/tex]:
- The slope [tex]\(m\)[/tex] is 2 and the y-intercept [tex]\(b\)[/tex] is -9.
- This means the line crosses the y-axis at (0, -9) and has a slope that rises 2 units for every 1 unit it moves to the right.
- For the second equation [tex]\( y = 2x - 3 \)[/tex]:
- The slope [tex]\(m\)[/tex] is 2 and the y-intercept [tex]\(b\)[/tex] is -3.
- This means the line crosses the y-axis at (0, -3) and also has a slope that rises 2 units for every 1 unit it moves to the right.
Step 3: Graph the lines.
1. Graph the first line [tex]\( y = 2x - 9 \)[/tex]:
- Start at the y-intercept (0, -9).
- Use the slope 2 to find another point by moving up 2 units and right 1 unit.
- Plot these points and draw the line.
2. Graph the second line [tex]\( y = 2x - 3 \)[/tex]:
- Start at the y-intercept (0, -3).
- Use the slope 2 to find another point by moving up 2 units and right 1 unit.
- Plot these points and draw the line.
Step 4: Analyze the graph.
When you graph both lines, you'll notice that they are parallel to each other. This is because they have the same slope of 2 but different y-intercepts (-9 and -3). Parallel lines do not intersect, which means there is no solution to the system of equations as there is no point [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously.
Conclusion:
The system of equations [tex]\( 2x - y = 9 \)[/tex] and [tex]\( 6x - 3y = 9 \)[/tex] has no solution. This corresponds to the situation where the lines are parallel and hence, the system is inconsistent.
