solve the syatem using the substitution method
4x - 5y = - 3
-x + 2y = 3
Using the substitution method means you first find one of the two variables (perhaps in terms of the other variable) and then 'substitute' this result in the other equation. If we apply it on this system we would work as follows:
1) Look at the second equation. You can easily rewrite this to:
x = 2y-3
So now you have x in terms of y
2) You now simple replace every x in the first equation with result you got from the second equation, meaning:
4x - 5y= -3
<=> 4(2y-3) - 5y = -3
<=> (8y - 12) - 5y = -3
<=> 3y = 9
<=> y = 3
3) Now that we have found that y equals 3, we can replace y with 3 in the second equation. We then find that:
x = 2y-3
<=> x = 2*3 - 3
<=> x = 3
So, the solution of this system is:
x=y=3
To solve the system using the substitution method, we need to isolate one variable in one of the equations and substitute it into the other equation.
Let's start with the second equation, which is:
-x + 2y = 3
We can isolate x by multiplying the equation by -1:
x - 2y = -3
Now, we can substitute this expression for x in the first equation:
4x - 5y = -3
Replacing x with (x - 2y), we get:
4(x - 2y) - 5y = -3
Expanding the brackets:
4x - 8y - 5y = -3
Combining like terms:
4x - 13y = -3
Now we have a new equation:
4x - 13y = -3
We can solve this equation for x. Let's isolate x:
4x = 13y - 3
Dividing both sides of the equation by 4:
x = (13y - 3) / 4
Now, we have an expression for x in terms of y. We can substitute this into one of the original equations to solve for y. Let's use the second equation:
-x + 2y = 3
Replacing x with [(13y - 3) / 4], we get:
-[(13y - 3) / 4] + 2y = 3
Multiplying through by 4 to clear the fraction:
-(13y - 3) + 8y = 12
Distributing the negative sign:
-13y + 3 + 8y = 12
Combining like terms:
-5y + 3 = 12
Subtracting 3 from both sides:
-5y = 9
Dividing both sides by -5:
y = -9/5
Now we have found the value of y. We can substitute this value back into the expression we found for x previously:
x = (13y - 3) / 4
Substituting y = -9/5:
x = (13(-9/5) - 3) / 4
Calculating:
x = (-117/5 - 15/5) / 4
x = (-132/5) / 4
Simplifying:
x = -33/5
So the solution to the system of equations is x = -33/5 and y = -9/5.