review the graphs of a system of two linear equations in two variables: y = x - 6 and y = -x +4 find the solution to both equations
The solution to both equations can be found at the point where the lines for each equation intersect on the graph.
Given the equations:
1) y = x - 6
2) y = -x + 4
You can set the equations equal to each other and solve for x:
x - 6 = -x + 4
By adding x to each side and adding 6 to each side, the solution becomes:
2x = 10
So, x = 10 / 2, therefore x = 5.
Substitute x = 5 into y = x - 6, y = 5 - 6 = -1.
The solution to the system of equations is the ordered pair (5, -1). Which represents the point of intersection of the two lines on the graph.
To review the graphs of the system of two linear equations, let's start by plotting each equation on a coordinate plane.
Equation 1: y = x - 6
To graph this equation, we need to find two points that satisfy it. We can choose arbitrary values of x and calculate the corresponding y values.
Let's choose x = 0:
y = 0 - 6
y = -6
So, we have the point (0, -6).
Let's choose x = 6:
y = 6 - 6
y = 0
So, we have the point (6, 0).
Plotting these two points on a graph, we get a line that passes through these points.
Equation 2: y = -x + 4
Similarly, we can find points that satisfy this equation.
Let's choose x = 0:
y = 0 + 4
y = 4
So, we have the point (0, 4).
Let's choose x = 2:
y = -2 + 4
y = 2
So, we have the point (2, 2).
By plotting these two points, we get another line on the graph.
Now, let's find the solution to both equations. The solution to the system of equations occurs at the point where the two lines intersect on the graph. By inspecting the graph, we can see that the point of intersection is (2, 2).
Therefore, the solution to both equations is x = 2 and y = 2.
To review the graphs of a system of two linear equations in two variables, we can start by plotting the equations on a coordinate plane.
The first equation is y = x - 6. To graph this equation, we need to find at least two points that satisfy this equation. We can choose any values for x and calculate the corresponding values for y. Let's choose x = 0 and x = 4 for simplicity.
When x = 0, y = 0 - 6 = -6. So we have the point (0, -6).
When x = 4, y = 4 - 6 = -2. So we have the point (4, -2).
Plotting these points on the coordinate plane and connecting them with a straight line gives us the graph of the first equation.
The second equation is y = -x + 4. Following the same process, we choose x = 0 and x = 4.
When x = 0, y = -0 + 4 = 4. So we have the point (0, 4).
When x = 4, y = -4 + 4 = 0. So we have the point (4, 0).
Plotting these points and connecting them with a straight line gives us the graph of the second equation.
Now, to find the solution to both equations, we need to find the point where the two lines intersect. In this case, it is the point (2, -4).
To solve for this point algebraically, we can set the two equations equal to each other and solve for x:
x - 6 = -x + 4
2x = 10
x = 5
Substituting x = 5 back into one of the equations, we find:
y = 5 - 6
y = -1
So the solution to both equations is (5, -1).
In summary, the solution to the system of equations y = x - 6 and y = -x + 4 is the point (5, -1), which is the intersection of the two graphs on the coordinate plane.