2x + 3 = -x + 5 - Coaching Toolbox
Understanding and Solving the Equation: 2x + 3 = -x + 5
Understanding and Solving the Equation: 2x + 3 = -x + 5
Solving linear equations is a fundamental skill in algebra, essential for students, educators, and professionals alike. One commonly encountered equation is 2x + 3 = -x + 5, which may seem simple at first glance but offers a great opportunity to reinforce algebraic thinking.
Understanding the Context
What Is the Equation 2x + 3 = -x + 5?
The equation 2x + 3 = -x + 5 sets two expressions equal to each other: on the left, 2 times a variable x plus 3; on the right, negative x plus 5. Solving this equation means finding the value of x that makes both sides equal. This process strengthens problem-solving and critical thinking skills.
Step-by-Step Solution
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Key Insights
Step 1: Eliminate the variable on one side
Start by moving all x terms to one side (here, the left) and constant terms to the other side (here, the right):
2x + 3 = -x + 5
2x + x + 3 = 5
2x + x = 3x, so the equation becomes:
3x + 3 = 5
Step 2: Isolate the x term
Subtract 3 from both sides to isolate the term with x:
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3x + 3 - 3 = 5 - 3
3x = 2
Step 3: Solve for x
Divide both sides by 3:
x = 2 ÷ 3
x = $\frac{2}{3}$
Verifying the Solution
Substitute $x = rac{2}{3}$ back into the original equation:
Left side:
2x + 3 = 2($\frac{2}{3}$) + 3 = $\frac{4}{3}$ + 3 = $\frac{4}{3}$ + $\frac{9}{3}$ = $\frac{13}{3}$
Right side:
-x + 5 = -\frac{2}{3} + 5 = -\frac{2}{3} + \frac{15}{3}$ = $\frac{13}{3}$
Both sides equal $ rac{13}{3}$, confirming the solution is correct.
