Q&A for How to Solve Equations with Variables on Both Sides

Solving for z, you would divide -54 by six to get singular z, with the end result being z=-9.

You solve for either variable in terms of the other variable (meaning that y will be equal to some expression with x in it, or that x will be equal to some expression with y in it). In this problem you could start by subtracting 9 from both sides of the equation, giving you 3y = 9x - 27. Then to solve for y, divide both sides of the equation by 3. To solve for x instead, first add 27 to both sides of the equation and then divide by 9.

Decide whether you want to solve for x in terms of y or vice versa. If solving for x, divide both sides of the equation by 9. That gives you (y/3) + 1 = x - 2. Add 2 to both sides: (y/3) + 3 = x. If solving for y, divide both sides by 3, which gives you y + 3 = 3x - 6. Subtract 3 from both sides: y = 3x - 9.

When you perform any operation on one side of an equation, you must perform exactly the same operation on the other side too.

You would perform a specific operation (add, subtract, multiply, or divide) to both sides of the equation in such a way that the variable would disappear from one side.

See Method 1 above.

Subtract 3x from both sides to get 7 = 14. Now there isn't a variable on either side! What value of x makes 7 = 14? None of them because 7 is never equal to 14, so your equation has no solutions.

First simplify as much as possible. 24x is already as simple as it can get. 48y / 1.6 simplifies to 30 y (by dividing 48 by 1.6). So 30 y = 24x. Simplify further by dividing both sides of the equation by 6: 5y = 4x. You still have the same equation you started with, but it's in a simpler form. Now you solve for one of the variables in terms of the other variable. To solve for x in terms of y, take the equation 5y = 4x, and divide both sides by 4, so that x = (5y) / 4. To solve for y in terms of x, divide both sides of 5y = 4x by 5, so that y = (4x) / 5. Both answers are correct. You can use either or both.

Multiply both sides of the equation by 32y (the product of the two denominators). Thus, (2/y)(32y) = (y/32)(32y). Perform the multiplications, so that 64 = y², and y = +/- 8. Either answer works.

Add g to both sides, and add 7/3 to both sides. That results in 1 + 7/3 = 3g, which simplifies to 10/3 = 3g. Divide both sides by 3: 10/9 = g.

Rewrite the equation substituting (a + 2) for c (since c is 2 greater than a). Thus, 2a + 15 = 3(a + 2) = 3a + 6. Subtract 2a from both sides, and subtract 6 from both sides: 9 = a, and c = 9 + 2 = 11.

First subtract 1 from both sides. 18 + 4x = -5x. Now subtract 4x from both sides: 18 = -9x. Finally, divide both sides by -9: -2 = x.

First subtract 1 from both sides: 18 + 4x = -5x. Now subtract 4x from both sides: 18 = -9x. Finally, divide both sides by -9: -2 = x.

Subtract 1 from both sides, and add 5x to both sides. You get 18 + 9x = 0. Then subtract 18 from both sides. 9x = -18, and x = -2.

When you have one equation in two unknowns, all you can do is solve for one variable in terms of the other. If 2b + 15 = 5a + b, subtract b from both sides, leaving b + 15 = 5a. Subtract 15 from both sides to find that b = 5a - 15. That's the only value for b we can get. If we return to b + 15 = 5a, and then divide both sides by 5, we find that a = (b + 15) / 5. Again, that's the only available value for a.

Add 9x to both sides. You get 24x - 10 = 2. Now add 10 on both sides: 24x = 12. Divide both sides by 24: x = 12/24 = 1/2.

If the variable is the same on both sides, the other numbers would have to be the same, too. The example equation you give is not valid, because if you subtract 2x from both sides, you're left with 3 = -4.

That's not true. Simplify both sides -2+1=-1, yet -4+9=5 so this would be a false statement and something you couldn't solve.

There's not a value that x is "mostly". Even if there was, you can't be 100% sure you know what the value is unless you solve it. But x is rarely zero, so you can eliminate zero if it is an option. However, this isn't recommended either, because you need to solve it to know for sure what the value of x is.

Multiply the denominator (x) to both sides; you'd get x+14=8. But then eliminate the x from the original numerator from both sides to get 14=7x. Divide both sides by 7, and you'll get an x value of 2.

It depends on the signs involved. If you multiply a negative by a positive, you’ll get a negative result, and if you multiply a negative by a negative, you’ll get a positive result.

The order of operations is PEMDAS: parentheses, exponents, multiply/divide, add/subtract. 12 ÷ 3 (5 - 1) = 12 ÷ 3 (4). Now you're left with nothing but multiplying and dividing, in which case you work from left to right: 12 ÷ 3 is 4, and (4)(4) = 16.

The idea is to isolate a variable on either side of an equation. It doesn't matter which side. You perform whatever operation(s) is necessary to achieve that isolation. For example, let's say you're solving this equation for x: 3x + 3 = 5x - 1. You want to wind up with 1x on either side of the equation. In this case the easiest thing to do is subtract 3x from both sides of the equation to wind up with no x on the left side and 2x on the right side. Then add 1 to both sides to wind up with +4 on the left side and no constant at all on the right side. So we have +4 = 2x. Finally, divide both sides by 2 in order to isolate x on the right side: 2 = x. You could have performed slightly different operations to isolate the x on the left side if you preferred.

Subtract 9x from both sides: 36 = 18 + x. Then subtract 18 from both sides: 18 = x.

8(7 + 3h) = 56 + 24h = 45. Subtract 56 on both sides: 24h = -11. So h = -11/24.

4y = (2y - 10) - 2 = 2y - 12. Subtract 2y from both sides. 2y = -12. So y = -6.

Subtract 3x from both sides: -24 = 2x. Divide both sides by 2: -12 = x.

Subtract 2x from both sides and add 5 to both sides. You get 5x = 20. Divide both sides by 5, leaving x = 4.

After simplifying, the first step in solving an equation with a variable on both sides is to get the variable on one side.

We need to manipulate the equation to get x by itself. 2(12(x-2))+2(3) = 2(1/2)(x+6)+2(2); 24(x-2)+6 = (x+6)+4 = x+10; 24x-48+6 = 24x-42 = x+6+4 = x+10; 24x-42 = x+10; 23x = 53; therefore, x = 53/23.