Solve the following system of linear equations by graphing. Graph on your own piece of paper. In your submitted answer, describe what the graph looks like and what this tells you about the solution to the system of linear equations.
y=x+3
−4x+4y=28
To graph the system of linear equations, we need to plot the points that satisfy the equations and then connect them to form the lines.
For the equation y=x+3, we can find several points that satisfy the equation. When x=0, y=3. When x=1, y=4, and so on. Plotting these points and connecting them will give us a line that starts at (0,3) and has a slope of 1 (rise of 1 and run of 1).
For the equation -4x+4y=28, we can rearrange it to y=x+7. Similarly, we can find points that satisfy this equation. When x=0, y=7. When x=1, y=8, and so on. Plotting these points and connecting them will give us a line that starts at (0,7) and has a slope of 1.
On the graph paper, we can plot these lines and see where they intersect. The point of intersection represents a solution to the system of linear equations.
The graph of the two lines will look like two straight lines intersecting at one point. This tells us that there is one unique solution to the system of linear equations. The point of intersection is the coordinates of this solution.
To solve the system of linear equations by graphing, we need to graph each equation on the same coordinate plane and find the point where the two lines intersect. This point of intersection represents the solution to the system of equations.
First, let's graph the equation y = x + 3:
To graph y = x + 3, we'll start with the y-intercept, which is 3. Plot the point (0, 3) on the coordinate plane. From this point, we can use the slope of 1 (since the coefficient of x is 1) to find additional points. The slope tells us that for every increase of 1 in the x-coordinate, the y-coordinate increases by 1.
For example, if we move to the right one unit from the y-intercept (0, 3), we would be at (1, 4). Similarly, if we move to the left one unit from the y-intercept, we would be at (-1, 2). We can continue this pattern to find more points.
Now, let's graph the equation -4x + 4y = 28:
To graph -4x + 4y = 28, we'll rearrange the equation to solve for y. Divide each term by 4: y = x + 7.
The y-intercept is 7, so plot the point (0, 7) on the coordinate plane. Again, we can use the slope of 1 to find additional points. Move one unit to the right from the y-intercept, and we are at (1, 8). Move one unit to the left from the y-intercept, and we are at (-1, 6).
Now, let's plot the points for both lines on the same graph.
The graph should show two lines. The line representing y = x + 3 is a straight line that goes upward to the right, starting from the point (0, 3). The line representing y = x + 7 is also a straight line that goes upward to the right, starting from the point (0, 7).
Now, observe the graph. If the lines intersect at a single point, then that point represents the solution to the system of linear equations. If the lines are parallel, they will not intersect and have no common solution. If the lines coincide, they are the same line and will have infinitely many solutions.
In this case, the two lines intersect at a single point (x, y). This indicates that the system of linear equations has a unique solution.
To find the coordinates of the point of intersection, you can visually estimate the coordinates, or if you have access to graphing software or a graphing calculator, you can use them to find the exact coordinates.
Once you have the coordinates of the point of intersection, you have found the solution to the system of linear equations.