Graph the system below and write its solution.
=3x+y−6
=y−1/2x-1
Graph the system below and write its solution.
=+−3xy−6
=y+−12x1
To graph the system of equations:
1. Set up a coordinate plane.
2. Solve each equation for y to find their slopes and y-intercepts.
3. Plot the y-intercepts on the coordinate plane.
4. Use the slopes to find additional points for each line and connect the points to graph the lines.
5. The solution will be the coordinates of the point (x, y) where the two lines intersect.
Equation 1: 3x + y - 6 = 0
Solving for y, we get:
y = -3x + 6
Equation 2: y - (1/2)x - 1 = 0
Solving for y, we get:
y = (1/2)x + 1
Now we can plot the lines on the coordinate plane.
Line 1: y = -3x + 6
- Find the y-intercept: (0, 6)
- Find additional points: Let x = 1, y = 3 → (1, 3)
Line 2: y = (1/2)x + 1
- Find the y-intercept: (0, 1)
- Find additional points: Let x = 2, y = 2 → (2, 2)
Plotting these points and connecting them with lines, we get:
To graph the system of equations and find its solution, we need to rewrite both equations in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
First equation:
3x + y − 6 = 0
Rearrange the equation to isolate y:
y = -3x + 6
Second equation:
y − 1/2x − 1 = 0
Rearrange the equation to isolate y:
y = 1/2x + 1
Now, we can graph the equations by plotting their corresponding slopes and y-intercepts:
First equation: y = -3x + 6
The slope is -3, which means for every unit increase in x, y decreases by 3.
The y-intercept is 6, which is the point where the line intersects the y-axis.
Plot the y-intercept (0, 6) on the graph. From there, use the slope to find other points. For example, if we move one unit to the right (x = 1), y will decrease by 3 from the y-intercept, giving us the point (1, 3). Similarly, if we move one unit to the left (x = -1), y will increase by 3 from the y-intercept, giving us the point (-1, 9). Connect these points to graph the line.
Second equation: y = 1/2x + 1
The slope is 1/2, which means for every unit increase in x, y increases by 1/2.
The y-intercept is 1, which is the point where the line intersects the y-axis.
Plot the y-intercept (0, 1) on the graph. From there, use the slope to find other points. For example, if we move two units to the right (x = 2), y will increase by 1 from the y-intercept, giving us the point (2, 1.5). Similarly, if we move two units to the left (x = -2), y will decrease by 1 from the y-intercept, giving us the point (-2, 0.5). Connect these points to graph the line.
Now that we have graphed both lines, we can determine their intersection, which represents the solution to the system of equations.
Looking at the graph, we can see that the two lines intersect at the point (2, 1.5). Therefore, the solution to the system of equations is x = 2 and y = 1.5.