The solution to a system of two linear equations is x = 3; y = 9.
How could the intersection of the graphs of the two equations be located on a coordinate grid?
To locate the intersection of the graphs of the two linear equations on a coordinate grid, follow these steps:
1. Write the two equations in slope-intercept form (y = mx + b), if they are not already in this form. For example, if the two equations are:
2x + 3y = 12
4x - y = 0
We can rearrange the first equation to get:
3y = -2x + 12
y = (-2/3)x + 4
And we can solve the second equation for y:
4x - y = 0
y = 4x
2. Plot the y-intercepts of each equation on the y-axis. The y-intercept is the value of y when x=0, so for the first equation (y = (-2/3)x + 4), the y-intercept is 4, and for the second equation (y = 4x), the y-intercept is 0. Plot these points on the y-axis.
3. Use the slope of each equation to find one or two additional points on each line. Remember that the slope is the "rise over run" or change in y divided by change in x. For example, for the first equation (y = (-2/3)x + 4), the slope is -2/3, so for each increase of 3 units in the x-direction, the y-coordinate will decrease by 2 units. Starting at the y-intercept of 4, we can go down 2 units and right 3 units to get another point on the line: (3,2). For the second equation (y = 4x), the slope is 4, so for each increase of 1 unit in the x-direction, the y-coordinate will increase by 4 units. Starting at the y-intercept of 0, we can go up 4 units and right 1 unit to get another point on the line: (1,4).
4. Plot the additional points on each line.
5. Find the intersection point of the two lines. This is the point where x and y satisfy both equations. We already know that x = 3 and y = 9 satisfy both equations, so this is the intersection point. Plot this point on the graph.
6. Check your work by verifying that the intersection point satisfies both equations. Plug in x=3 and y=9 into each equation to make sure that both sides are equal. If both sides are equal, then the point you found is indeed the intersection of the two lines.