Warm Up

A. Equation Solving

When converting units, it is useful to be able to solve equations like «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mi»x«/mi»«mo»=«/mo»«mn»15«/mn»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»t«/mi»«mn»4«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»5«/mn»«mn»12«/mn»«/mfrac»«/math».

Recall that solving an equation means finding a value for the variable that will make the equation true. A common method for solving equations is to apply operations to both sides of the equation to isolate the variable.

Example 1

Solve «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mi»x«/mi»«mo»=«/mo»«mn»15«/mn»«/math» for x.

 

Dividing both sides of the equation by 5 will isolate x on the left side of the equation.

Verify the solution by substituting x = 3 into the original equation to make sure the left side is equal to the right side.

Left Side Right Side
5 x
5(3)
15 		15
LS = RS


The left side equals the right side, so 3 is a solution to «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»5«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mn mathcolor=¨#B94A48¨»15«/mn»«/math».


Example 2

Solve «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»t«/mi»«mn»4«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»5«/mn»«mn»12«/mn»«/mfrac»«/math» for t.

If you prefer to use a decimal,  «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»5«/mn»«mn»12«/mn»«/mfrac»«mo»§#8729;«/mo»«mn»4«/mn»«/math» can be evaluated on a calculator as «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mo»§#247;«/mo»«mn»12«/mn»«mo»§#215;«/mo»«mn»4«/mn»«mo»=«/mo»«mn»1«/mn»«mo».«/mo»«mover»«mn»6«/mn»«mo»§#175;«/mo»«/mover»«/math» . The line on top of the 6 represents a «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mo».«/mo»«menclose notation=¨top¨»«mn»6«/mn»«/menclose»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»1«/mn»«mo».«/mo»«mn»6666«/mn»«mo».«/mo»«mo».«/mo»«mo».«/mo»«/math» Both «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mo».«/mo»«mover»«mn»6«/mn»«mo»§#175;«/mo»«/mover»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»5«/mn»«mn»3«/mn»«/mfrac»«/math» are acceptable answers.

Multiplying both sides of the equation by 4 will isolate t on the left side of the equation.


Again, the solution can be verified by substituting «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»t«/mi»«mo mathcolor=¨#B94A48¨»§#160;«/mo»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»§#160;«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo mathcolor=¨#B94A48¨».«/mo»«menclose mathcolor=¨#B94A48¨ notation=¨top¨»«mn»6«/mn»«/menclose»«mo mathcolor=¨#B94A48¨»§#160;«/mo»«/math» into the original equation to see if the left side is equal to the right side.

Left Side Right Side
LS = RS