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What steps are there to solving 3(2y + 7) = 10y + 9

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To solve the equation \(3(2y + 7) = 10y + 9\), follow these steps:

  1. Distribute on the left side: Apply the distributive property to eliminate the parentheses. \[ 3(2y) + 3(7) = 10y + 9 \] This simplifies to: \[ 6y + 21 = 10y + 9 \]

  2. Move all terms involving \(y\) to one side: Subtract \(6y\) from both sides of the equation to get all \(y\) terms on one side. \[ 21 = 10y - 6y + 9 \] Simplifying this, we have: \[ 21 = 4y + 9 \]

  3. Isolate the \(y\) term: Now, subtract 9 from both sides to isolate the term with \(y\). \[ 21 - 9 = 4y \] Simplifying this gives: \[ 12 = 4y \]

  4. Solve for \(y\): Finally, divide both sides by 4 to solve for \(y\). \[ y = \frac{12}{4} \] Simplifying this, we find: \[ y = 3 \]

So the solution to the equation \(3(2y + 7) = 10y + 9\) is \(y = 3\).

So what first? My answer choices are to put them in order
There's
Distribute the 3 to the parentheses
Subtract 6y to both sides
Divide 4 to both sides
Subtract 9 to both sides

To put the steps in order for solving the equation \( 3(2y + 7) = 10y + 9 \), here’s the correct sequence:

  1. Distribute the 3 to the parentheses:
    \( 3(2y + 7) \) becomes \( 6y + 21 \).

  2. Subtract 6y from both sides:
    This makes it \( 21 = 10y - 6y + 9 \) or \( 21 = 4y + 9 \).

  3. Subtract 9 from both sides:
    This leads to \( 12 = 4y \).

  4. Divide by 4 on both sides:
    Finally, this gives \( y = 3 \).

So the correct order of steps is:

  1. Distribute the 3 to the parentheses.
  2. Subtract 6y from both sides.
  3. Subtract 9 from both sides.
  4. Divide by 4 on both sides.
What does it mean for a linear equation to have infinity solutions?

When solving an equation, how can you tell it will have infinity many solutions?

A linear equation having infinitely many solutions means that there are countless values of the variable that satisfy the equation. This typically occurs when the equation represents the same line on a graph—meaning every point on the line is a solution to the equation.

Understanding Infinitely Many Solutions

  1. Identical Equations: If you manipulate the equation (for instance, through simplification or manipulation) and end up with a true statement that contains no variables (like \(0 = 0\)), it indicates that the original equations were essentially the same. For example:

    • If you start with \(2x + 4 = 8\) and simplify it to \(x + 2 = 4\), then to \(x = 2\), which has one solution. But if we had a situation where you simplify to a condition like \(0 = 0\), it indicates infinitely many solutions.

  2. Dependent Equations: If you have two equations that are dependent (one can be derived from another), they can represent the same line. For example:

    • If you have equations like \(2x + 4y = 8\) and \(4x + 8y = 16\), the second equation is a multiple of the first, meaning all points (x,y) that satisfy the first equation will also satisfy the second.

Recognizing Infinitely Many Solutions When Solving

  1. Manipulating the Equation: While solving a system of equations, if you reduce the equations and find:

    • An identity (like \(0 = 0\)).
    • Equations that are linear combinations of each other (one can be transformed into another through multiplication by a constant).

  2. Graphical Interpretation: If you graph two lines (or equations), and they overlap completely, then they represent the same line, which results in infinitely many points of intersection.

Example

Consider the equations:

  • Equation 1: \(2x + y = 4\)
  • Equation 2: \(4x + 2y = 8\)

If you simplify Equation 2 by dividing all terms by 2: \[ 2x + y = 4 \] This is exactly the same as Equation 1. Hence, both equations represent the same line, indicating that they have infinitely many solutions.

Summary

To conclude, a linear equation has infinitely many solutions when it represents the same entity as another equation in some form. You can identify this by manipulating the equations and checking for identities, or through graphical representation to see if the lines overlap.

What are 2 random Linear equations that both equal 0

Here are two random linear equations that equal 0:

  1. \(2x + 3y - 6 = 0\)

  2. \(4x - 5y + 10 = 0\)

Both of these equations can be rearranged to show that they equal zero.

Explanation:

  1. First Equation:

    • We can rewrite it to highlight that it equals 0: \[ 2x + 3y - 6 = 0 \]

  2. Second Equation:

    • Similarly, this one can be rewritten as: \[ 4x - 5y + 10 = 0 \]

Both of these equations represent lines in a 2-dimensional space, and you can solve or analyze them further if needed. If you graph them, they will intersect at some point unless they are parallel.