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Determine The Number of Solutions for Linear Equations
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 We'll learn that some algebraic equations have 1 solution.
 We’ll also learn that some algebraic equations have NO solution or INFINITE solutions.
 And we'll see how this knowledge can help us make delicious food, do science experiments, and even study animals!

Discussion Questions

Before VideoWhen modeling an equation with algebra tiles, how can you represent an equation you know is true?ANSWER

To model an equation we know is true, start with only ones tiles. Put the same number of ones on each side of the equation mat. We know that 2=2, for example, so if there are two ones on each side then the equation represented is true. Now any operations we do to both sides of the equation will continue to produce true equations.

To model an equation we know is untrue, start with only ones tiles. Put a different number of ones on either side of the equation mat. We know that 2=3, for example, is untrue, so if there are two ones on one side and three ones on the other, then the equation represented is untrue. Now, any operations we do to both sides will produce an equation that is untrue.

I noticed that my true equations always had the same number of xtiles and ones tiles on each side of the equation mat. Even when I added ones to both sides, or added xtiles to both sides, this stayed true.

I noticed that the untrue equations always had the same number of xtiles, but a different number of ones tiles on each side of the equation. Even when I added xtiles to both sides, or added ones tiles to both sides, this stayed true.

No, in order for an equation to be untrue the equation has to have the same number of xtiles on both sides, and a different number of ones tiles. Note that having no xtiles, means there are 0 xtiles on each side, which is still the same number of xtiles on each side.


After VideoWhat is the distributive property?ANSWER

The distributive property is a way of multiplying expressions that have more than one term in them. Every term in the first expression gets multiplied by every term in the second expression. For example, 3(x+4)=(3)(x)+(3)(4)=3x+12.

If there is one solution, one x is left in the equation. If the simplified equation has no variables left, then there are either no solutions or infinitely many solutions.

If you simplify an equation so that there are no variables left, and the equation you are left with is true, then there are infinitely many solutions. It means no matter what xvalue you substitute in, both sides of the equation are equal.

If you simplify the equation so that there are no variables left, and the equation you are left with is untrue, then there are no solutions. It means that no matter what value of x you substitute in, both sides of the equation are never equal.

If you can see that there are a different number of x’s on each side of the equation, then the equation has only one solution, and you can find it using inverse operations. For example, in the equation 3(x+3)=2x+9, use distribution on the left side to get 3x+9=2x+9. Since there are 3 x’s on the left and 2 x’s on the right, there is only one solution.



Vocabulary

Algebraic equation
DEFINE
An equation with a variable in it.

True equation
DEFINE
An equation where the value of the left side equals the value of the right side.

False equation
DEFINE
An equation where the value of the left side does not equal the value of the right side.

Variable
DEFINE
A letter that represents an unknown number.

Inverse operations
DEFINE
Operations that are opposites, so one operation undoes the other operation. Addition and subtraction are inverses, and multiplication and division are inverses.

Solution to an equation
DEFINE
Value(s) for the variable(s) in an algebraic equation that make(s) the equation true.

Distributive property
DEFINE
When a term is multiplied by a set of brackets that contain an expression, each term in the expression must be multiplied by the term outside the brackets. a(b+c)=ab+ac.

Algebraic equation
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageHOW MANY SOLUTIONS DOES AN ALGEBRAIC EQUATION HAVE?Algebraic equations can have 1 solution. They may also have infinitely many solutions, which means that any value of the variable makes the equation true. They may also have no solutions, which means that no value of the variable makes the equation true.
To better understand how to determine the number of solutions for linear equations …
HOW MANY SOLUTIONS DOES AN ALGEBRAIC EQUATION HAVE?. Algebraic equations can have 1 solution. They may also have infinitely many solutions, which means that any value of the variable makes the equation true. They may also have no solutions, which means that no value of the variable makes the equation true. To better understand how to determine the number of solutions for linear equations …LET’S BREAK IT DOWN!
Review how to solve an equation.
Emily wants to solve the equation 2x5=11. She knows she needs to do two steps to solve and use inverse operations to isolate the x. First, 5 is subtracted on the left side, so add 5 to both sides of the equation. 2x5+5=11+5. This simplifies to 2x=16. Next, the x is multiplied by 2, so divide both sides by 2. [ggfrac]2x/2[/ggfrac] = [ggfrac]16/2[/ggfrac]. This simplifies to x=8. x has been isolated, so you know that x is equal to 8. Try this yourself: Solve for x in the equation 3x+5=17.
Review how to solve an equation. Emily wants to solve the equation 2x5=11. She knows she needs to do two steps to solve and use inverse operations to isolate the x. First, 5 is subtracted on the left side, so add 5 to both sides of the equation. 2x5+5=11+5. This simplifies to 2x=16. Next, the x is multiplied by 2, so divide both sides by 2. [ggfrac]2x/2[/ggfrac] = [ggfrac]16/2[/ggfrac]. This simplifies to x=8. x has been isolated, so you know that x is equal to 8. Try this yourself: Solve for x in the equation 3x+5=17.Equations can have one solution.
Emily wants to solve the equation 5x8=3(x+4). This equation takes multiple steps to solve and has variables on both sides of the equation! On the right side, the 3 is multiplied by an expression in brackets. To simplify the right side, you need to use the distributive property. Multiply 3 by every term inside the brackets. 3(x + 4) = 3(x) + 3(4) = 3x + 12. Now the equation can be written as 5x8=3x+12. Isolate the xterms on one side of the equation (it doesn’t matter which side). Emily decides to put the xterms on the left side, so she focuses on eliminating the 8 first. Add 8 to both sides. 5x8+8=3x+12+8, which simplifies to 5x=3x+20. Next, subtract 3x from both sides. 5x3x=3x+203x, which simplifies to 2x=20. The last step to isolate the x is to divide both sides by 2. [ggfrac]2x/2[/ggfrac] = [ggfrac]20/2[/ggfrac]. Then x = 10. The equation has one solution, and that solution is 10. Try this yourself: Solve for x in 8x12=4(x+6).
Equations can have one solution. Emily wants to solve the equation 5x8=3(x+4). This equation takes multiple steps to solve and has variables on both sides of the equation! On the right side, the 3 is multiplied by an expression in brackets. To simplify the right side, you need to use the distributive property. Multiply 3 by every term inside the brackets. 3(x + 4) = 3(x) + 3(4) = 3x + 12. Now the equation can be written as 5x8=3x+12. Isolate the xterms on one side of the equation (it doesn’t matter which side). Emily decides to put the xterms on the left side, so she focuses on eliminating the 8 first. Add 8 to both sides. 5x8+8=3x+12+8, which simplifies to 5x=3x+20. Next, subtract 3x from both sides. 5x3x=3x+203x, which simplifies to 2x=20. The last step to isolate the x is to divide both sides by 2. [ggfrac]2x/2[/ggfrac] = [ggfrac]20/2[/ggfrac]. Then x = 10. The equation has one solution, and that solution is 10. Try this yourself: Solve for x in 8x12=4(x+6).Equations can have no solutions.
Adesina is solving 4x+8=4x+10. There is a variable on both sides of the equal sign, so move the variables to one side and the constants to the other. It doesn’t matter which you do first. First, let's remove the + 8 from the left side by subtracting 8 from both sides. 4x+88=4x+108, which simplifies to 4x=4x+2. Next, remove the 4x from the right side by subtracting 4x from both sides. 4x4x=4x+24x, or 0=2. This equation is not true! It does not matter what value of x you choose, you cannot make the left and right sides of the equation equal. This means that the equation has no solutions. Try this yourself: Show algebraically that 2x+4=2x+8 has no solutions.
Equations can have no solutions. Adesina is solving 4x+8=4x+10. There is a variable on both sides of the equal sign, so move the variables to one side and the constants to the other. It doesn’t matter which you do first. First, let's remove the + 8 from the left side by subtracting 8 from both sides. 4x+88=4x+108, which simplifies to 4x=4x+2. Next, remove the 4x from the right side by subtracting 4x from both sides. 4x4x=4x+24x, or 0=2. This equation is not true! It does not matter what value of x you choose, you cannot make the left and right sides of the equation equal. This means that the equation has no solutions. Try this yourself: Show algebraically that 2x+4=2x+8 has no solutions.Equations can have infinitely many solutions.
Emily is solving 12+3x=3x+12. Let's start by removing the 12 from the left side. So, subtract 12 from both sides: 12+3x12=3x+1212. When simplified this becomes 3x=3x. Next, remove the 3x from the right side by subtracting 3x from both sides. 3x3x=3x3x, which simplifies to 0=0. This statement is true. It doesn’t matter what value of x you choose, the left side is equal to the right side. This means that there are infinitely many solutions. Try this yourself: Show algebraically that 4+10x=2(5x+2) has infinitely many solutions.
Equations can have infinitely many solutions. Emily is solving 12+3x=3x+12. Let's start by removing the 12 from the left side. So, subtract 12 from both sides: 12+3x12=3x+1212. When simplified this becomes 3x=3x. Next, remove the 3x from the right side by subtracting 3x from both sides. 3x3x=3x3x, which simplifies to 0=0. This statement is true. It doesn’t matter what value of x you choose, the left side is equal to the right side. This means that there are infinitely many solutions. Try this yourself: Show algebraically that 4+10x=2(5x+2) has infinitely many solutions. 
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After simplifying a set of equations as much as possible, a student is left with (a) 2=2, (b) 4=0, and (c) x=3. How many solutions are there to each of the original equations?
Solve the equation 4x – 5 = 4x + 3. Identify the number of solutions.
Determine the number of solutions to the equation 4(2x+4)=8x+16.
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