Solve the system of equations graphically. y=−2x−1 3x - y =−7
To solve the system of equations graphically, we can plot the two linear equations on a graph and find the point of intersection.
First, let's plot the equation y = -2x - 1. We can start by plotting a few points on the graph and then connecting them to form a line:
When x = 0, y = -2(0) - 1 = -1. So one point on the line is (0, -1).
When x = 1, y = -2(1) - 1 = -3. So another point on the line is (1, -3).
When x = -1, y = -2(-1) - 1 = 1. So another point on the line is (-1, 1).
Plotting these three points on the graph and connecting them with a line, we get:
{{{graph(400,400,-5,5,-5,5,-2x-1)}}}
Next, let's plot the equation 3x - y = -7. To plot this line, we can again choose a few values for x and find the corresponding y-values:
When x = 0, 3(0) - y = -7, so y = 7. One point on the line is (0, 7).
When x = 1, 3(1) - y = -7, so y = 10. Another point on the line is (1, 10).
When x = -1, 3(-1) - y = -7, so y = -4. Another point on the line is (-1, -4).
Plotting these three points on the graph and connecting them with a line, we get:
{{{graph(400,400,-5,5,-5,5,-2x-1,3x+7)}}}
The point of intersection of these two lines is approximately (2, -5).
Therefore, the solution to the system of equations is x = 2 and y = -5.
To solve the system of equations graphically, let's first rearrange the equations into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Equation 1: y = -2x - 1
Equation 2: 3x - y = -7
For Equation 2, we need to solve for y:
y = 3x + 7
Now, we can plot the two equations on a graph to find their intersection point:
For Equation 1, the y-intercept is -1, and the slope is -2. This means that for every one unit increase in x, y will decrease by 2 units. We can plot a point at (0, -1) to start.
For Equation 2, the y-intercept is 7, and the slope is 3. This means that for every one unit increase in x, y will increase by 3 units. We can plot a point at (0, 7) to start.
Plotting the points and drawing the lines, we get:
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8 |- .
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______|__________________________________
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-1 |- .
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______|__________________________________
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0 2 4 6
From the graph, we can see that the two lines intersect at the point (2, 3). Therefore, the solution to the system of equations is x = 2 and y = 3.
To solve the system of equations graphically, we need to plot the graphs of the two equations on the same coordinate plane and find the point where they intersect. This point represents the solution to the system of equations.
First, let's plot the graph of the first equation, y = -2x - 1. To do this, we need to find at least two points that satisfy this equation.
Let's choose x = 0 and x = 2 as our x-values and substitute them into the equation to find the corresponding y-values:
For x = 0:
y = -2(0) - 1
y = 0 - 1
y = -1
So, one point on the graph of the first equation is (0, -1).
For x = 2:
y = -2(2) - 1
y = -4 - 1
y = -5
Another point on the graph of the first equation is (2, -5).
Plot these two points on the coordinate plane.
Next, let's plot the graph of the second equation, 3x - y = -7. Similarly, we need to find two points that satisfy this equation.
Again, let's choose x = 0 and x = 2 as our x-values and substitute them into the equation to find the corresponding y-values:
For x = 0:
3(0) - y = -7
0 - y = -7
-y = -7
y = 7
So, another point on the graph of the second equation is (0, 7).
For x = 2:
3(2) - y = -7
6 - y = -7
-y = -13
y = 13
One more point on the graph of the second equation is (2, 13).
Plot these two points on the same coordinate plane.
Now, we can observe the graph and notice that the two lines intersect at a single point. This point is the solution to the system of equations. In this case, it appears to be (-2, 3).
To check if this point is indeed the solution, substitute the x and y values into both equations:
For y = -2x - 1:
3 = -2(-2) - 1
3 = 4 - 1
3 = 3
The x and y values satisfy the first equation.
For 3x - y = -7:
3(-2) - 3 = -7
-6 - 3 = -7
-9 = -7
The x and y values also satisfy the second equation.
Therefore, the solution to the system of equations is (-2, 3).