Solving Equations With Fractions And Variables

Solving Equations With Fractions And Variables – There is no doubt that solving two-step equations is very easy. As the name suggests, two-step equations can be solved in only two steps.

Normally, when solving an equation, we always remember that what we do on one side of the equation must also be done on the other side for the equation to remain balanced.

Solving Equations With Fractions And Variables

Solving Equations With Fractions And Variables

We know that we have completely solved a two-phase equation if the variable, usually represented by a letter of the alphabet, is isolated on one side (left or right) of the equation and the number is on the opposite side.

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Note: This is the “normal” method as most two-phase equations are solved this way Note that step 2 can optionally be replaced with step 3 which is essentially the same

Solving Equations With Fractions And Variables

3) *Instead of step 2, always multiply both sides of the equation by the coefficient of the variable.

As the name of this linear equation suggests, solving for an unknown variable requires two steps. Usually the first step involves solving for the variable and getting rid of the “farthest” number from the term. Then we remove the “closest” number of variables. The number multiplies or divides the variable. It is also called the noise factor

Solving Equations With Fractions And Variables

Multi Step Equations With Fractions And Decimals

The variable here is [latex]x[/latex]. Our goal is to find [latex]x[/latex] by separating it on one side of the equation. Placing the variable on the left or right side doesn’t matter It’s up to you! This problem, let’s put it on the left as it is

On the side (left of the linear equation) where the variable is, note that [latex]2[/latex] is the “closest” to [latex]x[/latex] and [latex]5[/latex] is the “furthest”.

Solving Equations With Fractions And Variables

This simple observation allows us to decide which number to remove first. It’s definitely [latex]+5[/latex] since it’s the farthest between them. The opposite of [latex]+5[/latex] is [latex]-5[/latex], which means we multiply both sides of the equation by [latex]5[/latex].

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By [latex]5[/latex] subtracting both sides and constraining the left side of [latex]5[/latex], it’s time to join the closest or direct [latex]x[/latex] to [latex]2[/latex] in [latex]2[/latex]. Since [latex]2[/latex] multiplies [latex]x[/latex], the opposite is division by [latex]2[/latex].

Solving Equations With Fractions And Variables

After dividing both sides by [latex]2[/latex] we get the final answer or solution to the given two-phase linear equation.

Our goal is to keep [latex]x[/latex] on one side of the equation. It’s “standard” practice to solve for the left side of the variable Some algebra teachers may require you to put the variables on the left side, and nothing.

Solving Equations With Fractions And Variables

Linear Equations In One Variable

The first step removes the “farthest” number from the [latex]x[/latex] variable. Note that [latex]-3[/latex] is “closer” to [latex]x[/latex], while [latex]-8[/latex] is “farther away”. So we can eliminate [latex]-8[/latex] by adding its opposite, which is [latex]+8[/latex].

The second step is to get rid of the nearest integer in [latex]x[/latex], which is [latex]-3[/latex]. Since [latex]-3[/latex] multiplies x, the opposite is division by [latex]-3[/latex]. After dividing both sides by [latex]-3[/latex], we solved the linear equation.

Solving Equations With Fractions And Variables

The assumption is that we can isolate [latex]x[/latex] on the right side of the equation because it already exists.

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Looking at the right-hand side of the equation where the variable is located, [latex]3[/latex][latex]x[/latex] is the closest factor [latex]3[/latex] dividing the variable [latex]x[/latex]. On the other hand, the number [latex]26[/latex] is “further”. This means that by multiplying both sides of the equation by [latex]26[/latex], we get [latex]+26[/latex]. The reason we expand is that the additive inverse of [latex]+26[/latex] is [latex]-26[/latex].

Solving Equations With Fractions And Variables

The second step is to get rid of the name [latex]3[/latex]. Since [latex]3[/latex] divides [latex]x[/latex], the inverse is multiplied by [latex]3[/latex].

After multiplying both sides by [latex]3[/latex] we come to the final answer. You can rewrite your final answer as [latex]x = -9[/latex].

Solving Equations With Fractions And Variables

How To Solve Equations With Fractions — Krista King Math

This may look like a multi-step equation, but it’s not. It can be solved in two steps. Don’t worry about fractions, because it’s very easy. In this case, you’ll follow the rules for adding fractions. The rule says that if you add two fractions with the same name, just add the numbers and then copy the common name.

To solve the two-step equation above, to remove the fraction on the left-hand side, which is [latex]large}}[/latex], we’ll add [latex]large}}[/latex] to both sides of the equation.

Solving Equations With Fractions And Variables

Everything I mentioned above is just the first step. Now on to the second step. See the coefficients of the [latex]x[/latex] variable. It’s [latex]large}[/latex], which means the answer is [latex]large}[/latex].

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To finally solve a given equation, we multiply both sides of the equation by the coefficients of the given variable. Here is the full solution: Hi! Today we’re going to look at solving equations involving algebraic fractions But first let’s look at what algebraic fractions are An algebraic fraction is any fraction that contains an algebraic expression In other words, it’s a fraction that has a variable anywhere For example,

Solving Equations With Fractions And Variables

Now that we know what they are, let’s move on to the problem where we need to solve them.

It’s nice and simple We know that in order to solve an equation we need to isolate the variables Here we can see that our variable (x) is divisible by 8. To undo this division, just multiply both sides by 8

Solving Equations With Fractions And Variables

Solving Equations With Fractions And Decimals By

When we do that, our division is canceled and we are left with (x) and (7 times 8 = 56) on the left, so:

To solve this, I’ll first simplify my left-hand fraction. To do this, I need to convert (frac) to a fraction with a denominator of 12. I can do this by multiplying the number and the denominator by 3.

Solving Equations With Fractions And Variables

This gives me a fraction (frac). And then the rest of our equation will stay the same, so:

Solving One Step Equations With Fractional Coefficients

So if we look at it this way, the way to get rid of (frac) here is the inverse property. So the answer (frac) is (frac), so we multiply both sides.

Solving Equations With Fractions And Variables

When we do that, it deletes our fractions here, and we’re left with (x), which is what we want. And if we do it here, we can simplify by first doing (7div 7), which gives us 1, then (1times 12), which gives us 12.

Let’s do it a little differently For this problem, I’m going to solve (x) by eliminating fractions in our first step To do this, we multiply the whole equation by the least common multiple of the whole fractions in the problem

Solving Equations With Fractions And Variables

Question Video: Solving Simultaneous Equations By Elimination

In this case, our denominators are 5 and 10, and the least common multiple of 5 and 10 is 10. So multiply the whole equation by 10

This means we multiply each part of our equation by 10, which is (10cdot frac). We can do (10div 5) first because remember that it doesn’t matter if you multiply or divide first, the multiplication and division can be done at the same time. For this I will divide (10 div 5) and get 2, then (4x) to get (8x). If I multiply 10 by (frac), our 10s cancel out and we’re left with (3x). and (10 cdot 9 = 90).

Solving Equations With Fractions And Variables

Now we can solve this as a normal algebra problem. So I’ll subtract (3x) from both sides

A Fraction Becomes 9/11, If 2 Is Added To Both Numerator

Each of these methods will help you get the right answer when solving equations with algebraic fractions, so feel free to use whichever you feel most comfortable with.

Solving Equations With Fractions And Variables

So we have an expression in numbers, not just a variable, and maybe the number will be multiplied by that, so it’s a little different than what we’ve seen before. To solve such a problem, we first get rid of the fraction by multiplying the denominators on both sides of the equation. So we multiply both sides by 5

This cancels out our name, so we don’t have fractions anymore, and we’re left with a numeric expression: (x + 17). and (21 cdot 5 = 105).

Solving Equations With Fractions And Variables

How To Do Fractions, Equations And Algebra (the Easy Way)

And that’s all! I hope this video helped you better understand how to solve equations with algebraic fractions. Thanks for watching and have fun learning!

We can combine fractions on the left side by converting (frac) to a fraction

Solving Equations With Fractions And Variables

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