Classify each of the following equations. 6 ( x − 1 ) + 13 − x = 5 x + 7 6\left(x-1\right)+13-x=5x+7

Welcome back. For this problem, we'd like to classify the following equation, six times x minus one plus 13 minus x equals five x plus seven. To classify this equation, we'll have to solve it to see how many solutions it has, so let's get into it. We need to start by simplifying the equation. On the left side, we have to distribute our six, so let's start there. We have six times x, which is six x, and then six times minus one, which is minus six. We still have our plus 13 and our minus x, and then on the right side of the equation, we still have five x plus seven. Now we need to combine like terms. We have six x minus x, which is five x, And then we have minus six plus 13, which is plus seven. So we get five x plus seven equals five x plus seven. Maybe we can see where this is going, but let's carry it through just to be 100% sure. We wanna get all of our constants on one side of the equation and all of our variable terms on the other side of the equation. And we currently have variable and constant terms on both sides, so let's try to get variables to the left, constants to the right, starting with our constants. We wanna get rid of this plus seven on the left, which we can do by subtracting seven on both sides of the equation. It will cancel the plus seven on the left, leaving us with just five x, and it will also cancel the plus seven on the right, again, leaving us with just five x. Next, we want to get rid of this five x on the right side of the equation, which we can do by subtracting five x on both sides of the equation. It will cancel the five x on the right, leaving us with zero, and it will also cancel the five x on the left, again leaving us with zero. So we get the statement zero equals zero, which is always a true statement. So we can see that we could plug in any of the infinite real numbers for x, and we would end up getting a true statement. We have infinitely many solutions, this statement is always true, so we can say that this equation is an identity equation. Great work.

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Multiple Choice

Classify each of the following equations.
$6 (x - 1) + 13 - x = 5 x + 7$

Conditional

Identity

Contradiction

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Start by writing down the given equation: $6\left(x-1\right) + 13 - x = 5x + 7$.

Distribute the 6 on the left side to both terms inside the parentheses: $6 \times x$ and $6 \times (-1)$, which gives \$6x - 6$.

Rewrite the equation after distribution: \$6x - 6 + 13 - x = 5x + 7$.

Combine like terms on the left side: combine \$6x$ and $-x$ to get $5x$, and combine $-6$ and $13$ to get $7$, so the left side becomes $5x + 7$.

Now the equation looks like \$5x + 7 = 5x + 7$. Since both sides are identical expressions, this means the equation is true for all values of $x$, which classifies it as an identity.

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Classify each of the following equations.
$6 (x - 1) + 13 - x = 5 x + 7$