Question

solve the equation and check the solution.
10x + 4x + 3 + 9 = 9x + 3 + 4x

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution is x = \square. (type an integer or a decimal.)
b. the solution is all real numbers.
c. there is no solution.

Explanation:

Step1: Combine like terms

Left side: \(10x + 4x + 3 + 9 = 14x + 12\)
Right side: \(9x + 3 + 4x = 13x + 3\)
Equation becomes: \(14x + 12 = 13x + 3\)

Step2: Subtract \(13x\) from both sides

\(14x - 13x + 12 = 13x - 13x + 3\)
Simplify: \(x + 12 = 3\)

Step3: Subtract 12 from both sides

\(x + 12 - 12 = 3 - 12\)
Simplify: \(x = -9\) Wait, no, wait, let's re - check. Wait, original left side: \(10x + 4x=14x\), \(3 + 9 = 12\). Right side: \(9x+4x = 13x\), \(+3\). So equation is \(14x + 12=13x + 3\). Subtract \(13x\): \(x+12 = 3\). Subtract 12: \(x=3 - 12=-9\). Wait, but let's check the original equation again. Wait, maybe I made a mistake. Wait, original equation: \(10x + 4x+3 + 9=9x + 3+4x\). Let's re - combine:

Left: \(14x+12\), Right: \(13x + 3\). Then \(14x+12-(13x + 3)=14x + 12-13x - 3=x + 9\). Wait, no, \(14x+12 = 13x+3\) => \(14x-13x=3 - 12\) => \(x=-9\). Wait, but let's plug \(x = - 9\) back into the original equation.

Left side: \(10(-9)+4(-9)+3 + 9=-90-36 + 12=-114\)

Right side: \(9(-9)+3+4(-9)=-81 + 3-36=-114\). So it works. Wait, but wait, maybe I misread the equation. Wait, the original equation is \(10x + 4x+3 + 9=9x + 3+4x\). Wait, another way: combine like terms again.

Left: \(14x+12\), Right: \(13x + 3\). Subtract \(13x\) and 3 from both sides: \(14x-13x+12 - 3=13x-13x+3 - 3\) => \(x + 9=0\) => \(x=-9\). So the solution is \(x=-9\). Wait, but the options have A: solution is \(x=\), B: all real numbers, C: no solution. So the correct answer is A, with \(x=-9\).

Wait, let's re - do the steps carefully:

1. Combine like terms on the left - hand side (LHS) and right - hand side (RHS) of the equation \(10x + 4x+3 + 9=9x + 3+4x\)

• LHS: \(10x+4x = 14x\), \(3 + 9=12\), so LHS is \(14x + 12\)
• RHS: \(9x + 4x=13x\), so RHS is \(13x + 3\)
• The equation becomes \(14x+12 = 13x + 3\)

1. Isolate the variable \(x\)

• Subtract \(13x\) from both sides: \(14x-13x+12=13x - 13x+3\)
• Simplify: \(x + 12=3\)
• Subtract 12 from both sides: \(x=3 - 12=-9\)

Answer:

A. The solution is \(x=-9\)