hope this helps โ
MULTIPLY THE FIRST EQUATION BY -2 & ADD THEM.
-4X+8Y=-14
4X-2Y=9
6Y=-5
Y=-5/6 ANSWER.
4X-2*-5/6=9
4X+10/6=9
Step-by-step explanation:
Lets start with the given system of linear equations
2%2Ax-4%2Ay=7
4%2Ax-2%2Ay=9
In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).
So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.
So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 2 and 4 to some equal number, we could try to get them to the LCM.
Since the LCM of 2 and 4 is 4, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:
2%2A%282%2Ax-4%2Ay%29=%287%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%284%2Ax-2%2Ay%29=%289%29%2A-1 Multiply the bottom equation (both sides) by -1
So after multiplying we get this:
4%2Ax-8%2Ay=14
-4%2Ax%2B2%2Ay=-9
Notice how 4 and -4 add to zero (ie 4%2B-4=0)
Now add the equations together. In order to add 2 equations, group like terms and combine them
%284%2Ax-4%2Ax%29-8%2Ay%2B2%2Ay%29=14-9
%284-4%29%2Ax-8%2B2%29y=14-9
cross%284%2B-4%29%2Ax%2B%28-8%2B2%29%2Ay=14-9 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.
So after adding and canceling out the x terms we're left with:
-6%2Ay=5
y=5%2F-6 Divide both sides by -6 to solve for y
y=-5%2F6 Reduce
Now plug this answer into the top equation 2%2Ax-4%2Ay=7 to solve for x
2%2Ax-4%28-5%2F6%29=7 Plug in y=-5%2F6
2%2Ax%2B20%2F6=7 Multiply
2%2Ax%2B10%2F3=7 Reduce
2%2Ax=7-10%2F3 Subtract 10%2F3 from both sides
2%2Ax=21%2F3-10%2F3 Make 7 into a fraction with a denominator of 3
2%2Ax=11%2F3 Combine the terms on the right side
cross%28%281%2F2%29%282%29%29%2Ax=%2811%2F3%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.
x=11%2F6 Multiply the terms on the right side
So our answer is
x=11%2F6, y=-5%2F6
which also looks like
(11%2F6, -5%2F6)
Notice if we graph the equations (if you need help with graphing, check out this solver)
2%2Ax-4%2Ay=7
4%2Ax-2%2Ay=9
we get
graph of 2%2Ax-4%2Ay=7 (red) 4%2Ax-2%2Ay=9 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).
and we can see that the two equations intersect at (11%2F6,-5%2F6). This verifies our answer.You really don't want to solve this system by 'addition.' What you want is to solve it by elimination, using addition as one of the steps of the process.
The idea of solving by elimination is to multiply (if necessary) one of the equations by some constant so that one of the variables will have a coefficient that is the additive inverse of the coefficient on the same variable in the other equation.
In this example, multiply the first equation by -2, resulting in a coefficient on the x term of -4. -4 is the additive inverse of the 4 coefficient on x in the second equation.
-4x+%2B+8y+=+-14
4x-2y=9
Now you can add the two equations, term-by-term. That is, you add the x terms to the x terms, the y terms to the y terms, and the constants to the constants, resulting in one equation with a zero coefficient on one of the variables, in this case, x.
The sum equation is:
0x%2B6y=-5
Divide by 6:
y=-%285%2F6%29
Now that you have a value for y, you can either substitute that value into either one of the original equations and solve for x, or you could take the original two equations, multiply one of them by an appropriate constant to eliminate the y variable so that you can solve for x. Either way works. Watch carefully:
4x-2%28-5%2F6%29=9
4x%2B5%2F3=9
4x=%2827%2F3%29-%285%2F3%29
4x=22%2F3
x=11%2F6
