x+3y=1
-5x+4y=-24
Solve the linear system by using elimination
To solve the linear system using elimination, we need to eliminate one variable by adding or subtracting the equations. Let's start by eliminating the x variable.
We can multiply the first equation by 5 and the second equation by 1 to make the coefficients of x in both equations the same:
5(x + 3y) = 5(1)
1(-5x + 4y) = 1(-24)
This simplifies to:
5x + 15y = 5
-5x + 4y = -24
Now we can add the two equations together:
(5x + 15y) + (-5x + 4y) = 5 + (-24)
Simplifying this equation gives:
19y = -19
Dividing both sides of the equation by 19 yields:
y = -1
Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use the first equation:
x + 3(-1) = 1
Simplifying this equation gives:
x - 3 = 1
Adding 3 to both sides of the equation yields:
x = 4
Therefore, the solution to the linear system is x = 4 and y = -1.
To solve the linear system using the elimination method, follow these steps:
Step 1: Multiply one or both equations by a constant to make the coefficients of either 'x' or 'y' in both equations the same or additive inverses.
In this case, let's start by making the coefficient of 'x' the same in both equations. Multiply the first equation by 5 and the second equation by -1 to achieve this:
5(x + 3y) = 5(1) ==> 5x + 15y = 5 (equation 1)
-1(-5x + 4y) = -1(-24) ==> 5x - 4y = 24 (equation 2)
Step 2: Add the two modified equations together to eliminate the 'x' variable.
(5x + 15y) + (5x - 4y) = 5 + 24
When we combine like terms, we get:
10x + 11y = 29
Step 3: Now you have a new equation with only 'y'. Solve for 'y' by performing inverse operations.
To isolate 'y', subtract 10x from both sides:
10x + 11y - 10x = 29 - 10x
This simplifies to:
11y = 29 - 10x
Step 4: Divide both sides of the equation by 11 to solve for 'y':
11y/11 = (29 - 10x)/11
y = (29 - 10x)/11
Step 5: Substitute the value of 'y' back into either of the original equations to solve for 'x'.
Let's substitute the value of 'y' into the first equation:
x + 3((29 - 10x)/11) = 1
Now, solve for 'x'.
Multiply both sides of the equation by 11 to eliminate the fraction:
11x + 90 - 30x = 11
Combine like terms:
-19x + 90 = 11
Subtract 90 from both sides:
-19x = 11 - 90
-19x = -79
Finally, divide both sides by -19 to solve for 'x':
x = -79/-19
x = 4.16 (rounded to two decimal places)
Step 6: Substitute the value of 'x' back into either of the original equations to solve for 'y'.
Let's substitute 'x' into the second equation:
-5(4.16) + 4y = -24
Now, solve for 'y'.
-20.8 + 4y = -24
Add 20.8 to both sides:
4y = -24 + 20.8
4y = -3.2
Divide both sides by 4 to solve for 'y':
y = -3.2/4
y = -0.8
Therefore, the solution to the linear system is x = 4.16 and y = -0.8.
#1 times 5 --->
5x + 15y = 5
add to the 2nd
19y = -19
y = -1
sub into x+3y=1
x-3 = 1
x = 4
x=4, y = -1