Review the graphs of a system of these two linear equations in two variables: y= 4x -10 and y = -2x + 8 Find the solution of the system.
To find the solution to a system of two linear equations, we can either solve it graphically or algebraically.
Graphically:
If we graph both equations on the same set of axes, where each axis represents one variable, the point where the two lines intersect is the solution to the system.
For y = 4x - 10, this is a line with a slope of 4 and a y-intercept at (0, -10). This means the line goes up 4 units for every 1 unit it moves to the right.
For y = -2x + 8, this is a line with a slope of -2 and a y-intercept at (0, 8). This means the line goes down 2 units for every 1 unit it moves to the right.
If you were to draw these two lines on graph paper, they would intersect at a point, and that point would be the solution to the system. However, because we are not actually drawing the graph, we'll solve for it algebraically instead.
Algebraically:
To solve the system algebraically, we can set the two equations equal to each other, since y is equal to both 4x - 10 and -2x + 8.
So, setting the two right-hand sides of the equations equal to each other:
4x - 10 = -2x + 8
Next, we move like terms to each side of the equation to solve for x:
4x + 2x = 8 + 10
This simplifies to:
6x = 18
Dividing both sides by 6 gives us:
x = 18 / 6
So, x = 3
Now that we have found x, we can substitute it back into either of the original equations to find the value of y. Let's use the first equation for this example:
y = 4x - 10
Substitute x with 3:
y = 4(3) - 10
y = 12 - 10
y = 2
Therefore, the solution to the system of equations is x = 3 and y = 2. If you were to graph both equations, the lines would intersect at the point (3, 2).