Solve each linear system. Verify each solution by substituting the coordinates of your solutino into both equations.
a) y = -x and y = x - 6
b) x - y = 8 and x + 2y = 2
well, do what they said
y = x - 6
but they said y = -x
so
-x = x-6
6 = 2 x
x = 3
in the next one
y = (x-8)
use that in the second equation
To solve a linear system, we need to find the values of the variables that make both equations true at the same time. We can use substitution or elimination method to solve these systems.
a) y = -x and y = x - 6:
We can see that both equations are already solved for y. So, we can substitute one equation into the other to find the value of x.
Substituting y = -x into the second equation:
-x = x - 6
Now, we have a linear equation with one variable:
2x = 6
Divide both sides by 2:
x = 3
To find the value of y, substitute the value of x into any of the given equations:
y = -x
y = -3
So, the solution to this system is x = 3 and y = -3.
To verify this solution, substitute the values of x and y into both equations:
Checking the first equation: y = -x
-3 = -3 (True)
Checking the second equation: y = x - 6
-3 = 3 - 6
-3 = -3 (True)
Both equations hold true when the values of x = 3 and y = -3, confirming the solution.
b) x - y = 8 and x + 2y = 2:
In this case, the equations are not already solved for one variable, so we will use the elimination method by adding or subtracting the equations to eliminate one variable.
Multiply the first equation by 2 to make the coefficients of y the same:
2(x - y) = 2(8)
2x - 2y = 16
Now, add this equation to the second equation:
2x - 2y + x + 2y = 16 + 2
3x = 18
Divide both sides by 3:
x = 6
To find the value of y, substitute the value of x into any of the given equations:
x - y = 8
6 - y = 8
Subtract 6 from both sides:
-y = 2
Multiply both sides by -1:
y = -2
So, the solution to this system is x = 6 and y = -2.
To verify this solution, substitute the values of x and y into both equations:
Checking the first equation: x - y = 8
6 - (-2) = 8
8 = 8 (True)
Checking the second equation: x + 2y = 2
6 + 2(-2) = 2
6 - 4 = 2
2 = 2 (True)
Both equations hold true when the values of x = 6 and y = -2, confirming the solution.