A solution to an equation is any value that can replace the variable and make a true statement. To determine whether or not a value satisfies an equation, we need to check whether the left-hand side of the equation is equal to the right-hand side of the equation. However, an equation may not have just one answer.
Complete the table of values by evaluating each side of equation 1:
3x -5 = 3x-5
x | 3x-5 | 3x-5 |
---|---|---|
-1 | ||
0 | ||
1 | ||
2 | ||
3 |
Now, complete the table for equation 2:
9+x=x+5
x | 9+x | x+5 |
---|---|---|
-1 | ||
0 | ||
1 | ||
2 | ||
3 |
Finally, complete the table for equation 3:
35+5x=2x+35
x | 35+5x | 2x+35 |
---|---|---|
-1 | ||
0 | ||
1 | ||
2 | ||
3 |
How many solutions do you think there will be to each equation? Why?
How does the structure of an equation relate to the number of solutions it has?
We can determine the solutions to an equation by looking at its structure. To illustrate the difference, we will show how equations look different depending on having one solution, no solution, or infinitely many solutions using the starting equation
3(x+2)=â¬š
If the equation simplifies to variables with the same coefficients and the same constants on each side of the equality, it will have infinitely many solutions.
The equation represented is 3(x+2)=x+2(x+3), which has infinite solutions.
When each side is simplified, the equation becomes 3x+6=3x+6, shown on the balance by three +x tiles and six +1 tiles.
Regardless of what value of x is chosen for the +x tile, there are equal numbers of +x tiles on each side that will be added to the constants.
The number of +1 tiles are also equal on each side. This creates a balanced scale regardless of the value chosen for x.
If the equation simplifies to variables with the same coefficients but different constants, it will have no solution.
If the equation simplifies to variables with different coefficients, there will be a unique solution regardless of constant values.
While there is more than one way to algebraically solve an equation, it is important to keep both sides of an equation equivalent by remembering that what you do to one side, you must do to the other.
Verify that the m=-4 is a solution to the equation 14-7m= -3(-2+3m).
Use a balance scale and algebra tiles to solve 3(x-4)=2(-2x+1).
Determine how many solutions the following equations have without solving.
4\left(x-9\right)=x+6
2x-5=0.5\left(4x+10\right)
Solve the following equations:
4\left(5x+1\right)=-3\left(5x-5\right)
\dfrac{\left(x+3\right)}{2}+1.3x=\dfrac{0.8x}{4}
Right now, Bianca's father is 48 years older than Bianca.
Two years ago, her father was 5 times older than her.
Solve for y, Bianca's current age.
A solution to an equation is any value that can replace the variable and make a true statement.
A fully simplified equation in one variable will take one of the following three forms, corresponding to how many solutions the equation has:
x=a, where a is a number (a unique solution)
a=a, where a is a number (infinitely many solutions)
a=b, where a and b are different numbers (no solutions)
We can simplify an equation using the distributive property or the multiplication property of equality to eliminate fractions or decimals. After simplifying, we can continue using properties of equality to solve for the unknown.