Question

1. here is an equation: 2x + 9 = -15. write three different equations that have the same solution as 2x + 9 = -15. show or explain how you found them.

Explanation:

Step1: Solve the original equation

First, solve \(2x + 9=-15\). Subtract 9 from both sides: \(2x=-15 - 9=-24\), then divide by 2: \(x = - 12\).

Step2: Create first new equation

We can multiply both sides of the original equation by a non - zero number, say 2. So \(2(2x + 9)=2\times(-15)\), which simplifies to \(4x+18=-30\). When we solve this equation: subtract 18 from both sides, \(4x=-30 - 18=-48\), then \(x=\frac{-48}{4}=-12\).

Step3: Create second new equation

Add the same number to both sides of the original equation. Let's add 5 to both sides. So \((2x + 9)+5=-15 + 5\), which simplifies to \(2x+14=-10\). Solving this equation: subtract 14 from both sides, \(2x=-10 - 14=-24\), then \(x=\frac{-24}{2}=-12\).

Step4: Create third new equation

We can rewrite the original equation by isolating \(x\) in a different way. From \(2x+9=-15\), we can also write it as \(2x=-15 - 9\), or \(2x=-24\). Another way is to divide both sides of \(2x=-24\) by 2 first and then add 3 to both sides. Divide \(2x=-24\) by 2: \(x=-12\), then add 3 to both sides: \(x + 3=-12+3=-9\), or \(x=-9\) (but we want an equation with \(x\) on one side and an expression on the other, so we can multiply both sides by 1 (which doesn't change it) and write \(x+3=-9\). Solving \(x + 3=-9\): subtract 3 from both sides, \(x=-9 - 3=-12\).

Answer:

1. \(4x + 18=-30\) (obtained by multiplying both sides of \(2x + 9=-15\) by 2)
2. \(2x+14=-10\) (obtained by adding 5 to both sides of \(2x + 9=-15\))
3. \(x+3=-9\) (obtained by a series of equivalent transformations to keep the solution \(x = - 12\))