Equations with Brackets and Unknowns on Both Sides

Subtract the same x-term from both sides to collect the unknowns.

Remove the smaller x-term first, then undo the constant and multiplier.

After collecting the x-terms, undo subtracting 4.

The whole bracket is multiplied by 2; you can undo that first.

After dividing, use the operation that undoes the subtraction inside the bracket.

Expand the bracket, multiplying both terms by 4.

1. Expand: 4x+4=2x+14.
2. Subtract 2x and then 4: 2x=10.
3. Divide by 2: x=5. Check: both original sides equal 24.

It is convenient to keep the larger x-term on the right.

1. Subtract 2x from both sides: 11=3x+2.
2. Subtract 2: 9=3x. Divide by 3: x=3.
3. Check: 2\times3+11=17 and 5\times3+2=17.

A negative solution is allowed; keep the sign when dividing.

1. Subtract 3x: 2x+4=-8.
2. Subtract 4: 2x=-12. Divide by 2: x=-6.
3. Check: both original sides equal -26.

After expanding, subtract 2x from both sides.

1. Expand: 2x+8=3x+3.
2. Subtract 2x: 8=x+3. Subtract 3: x=5.
3. Check: 2(5+4)=18 and 3\times5+3=18.

Combine the number terms after expanding, before collecting the x-terms.

1. Expand and simplify: 3x-3+2=2x+5, so 3x-1=2x+5.
2. Subtract 2x: x-1=5. Add 1: x=6.
3. Check: 3(6-1)+2=17 and 2\times6+5=17.

Write both expanded sides before simplifying.

1. Expand both brackets: 4x-8=2x+10.
2. Subtract 2x: 2x-8=10. Add 8: 2x=18.
3. Divide by 2: x=9. Check: 4(9-2)=28 and 2(9+5)=28.

Write an expression for each answer, then set the expressions equal.

1. Let the starting number be x. The equal answers give 3x+7=5x+2.
2. Subtract 3x from both sides: 7=2x+2.
3. Subtract 2: 5=2x. Divide by 2: x=2.5.
4. Check: 3\times2.5+7=14.5 and 5\times2.5+2=14.5.

Adding 2x to both sides removes the negative x-term.

1. Add 2x to both sides: 18=5x-7.
2. Add 7: 25=5x. Divide by 5: x=5.
3. Check: 18-2\times5=8 and 3\times5-7=8.

Write an expression for each area. Use brackets when multiplying a side with two terms.

1. The first area is 6(x+2). The second area is 4(2x-1).
2. The areas are equal, so 6(x+2)=4(2x-1).
3. Expand: 6x+12=8x-4. Subtract 6x and add 4: 16=2x.
4. Divide by 2: x=8.
5. Check: the lengths are 10 and 15 cm. Both areas are 60\text{ cm}^2, since 6\times10=4\times15=60.

The x-term inside the bracket is negative; keep it negative when multiplying.

1. Expand carefully: 10-2x=x+1.
2. Add 2x: 10=3x+1. Subtract 1: 9=3x.
3. Divide by 3: x=3. Check: 2(5-3)=4 and 3+1=4.

For (a), multiply every term inside the bracket by 4. For (b), use the corrected equation.

Part (a)

1. The 4 multiplies both x and -3.
2. Since 4\times(-3)=-12, the expanded left side is 4x-12.

Part (b)

1. Use 4x-12=2x+6. Subtract 2x: 2x-12=6.
2. Add 12: 2x=18. Divide by 2: x=9.
3. Check: 4(9-3)=24 and 2\times9+6=24.

For (a), write each complete perimeter and set them equal. For (b), substitute your value into the side lengths.

Part (a)

1. The square’s perimeter is 4(x+1). The rectangle’s perimeter is 2(x+2)+2\times5=2x+14.
2. Set them equal: 4x+4=2x+14. Subtract 2x and 4 to get 2x=10.
3. So x=5.

Part (b)

1. The square’s side is 5+1=6 cm, so its perimeter is 4\times6=24 cm.
2. Check the rectangle: length 5+2=7 cm, width 5 cm, giving 2\times7+2\times5=24 cm.

Use one expression for each total cost and set them equal.

1. Let d be the days. Equal costs give 12+3d=5+4.40d.
2. Subtract 3d and then 5: 7=1.40d.
3. Divide by 1.40: d=5 days.
4. Check: A costs 12+3\times5=27 pounds; B costs 5+4.40\times5=27 pounds.

For (a), remove the same number of boxes from each side of the balance. For (b), include both the box and its contents.

Part (a)

1. Let an empty box have mass b grams. Then 3b+400=5b+100.
2. Subtract 3b and then 100: 300=2b. Divide by 2: b=150 g.
3. Check: both sides total 850 g.

Part (b)

1. Use the empty box mass from part (a) and add the beads.
2. 150+250=400 g.

Let the daughter's age be x. Write the mother's age in two different ways, then make them equal.

Part (a)

1. Let the daughter’s age be x. The mother’s age is 3x, and it is also x+24.
2. So 3x=x+24. Subtract x from both sides: 2x=24.
3. Divide by 2: x=12. The daughter is 12 years old.

Part (b)

1. The mother is three times the daughter’s age: 3\times12=36 years.
2. Check: 36-12=24, so she is 24 years older, as stated.