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Convert $y = \\frac{1}{2}(x - 4) + 3$ to Slope-Intercept Form (Algebra 1)
Mathematics
High School Grade 9 (Algebra 1)
Question Content
Put the equation $y = \\frac{1}{2}(x - 4) + 3$ in Slope Intercept form. Show all steps Algebraically.
Correct Answer
$y = \\frac{1}{2}x + 1$
Detailed Solution Steps
1
Step 1: Recall slope-intercept form is $y = mx + b$. Start by expanding the right-hand side using the distributive property.
2
Step 2: Distribute $\\frac{1}{2}$ to the terms inside the parentheses: $y = \\frac{1}{2}x - \\frac{1}{2}(4) + 3$.
3
Step 3: Calculate $\\frac{1}{2}(4) = 2$, so the equation becomes $y = \\frac{1}{2}x - 2 + 3$.
4
Step 4: Combine the constant terms: $-2 + 3 = 1$, resulting in the final slope-intercept form: $y = \\frac{1}{2}x + 1$.
Knowledge Points Involved
1
Slope-Intercept Form of a Linear Equation
The slope-intercept form is written as $y = mx + b$, where $m$ represents the slope (rate of change of the line) and $b$ represents the y-intercept (the point where the line crosses the y-axis). This form is used to easily graph linear equations and identify key characteristics of the line.
2
Distributive Property
The distributive property states that $a(b + c) = ab + ac$. It is used to expand expressions with parentheses, which is necessary to convert point-slope form to slope-intercept form.
3
Combining Like Terms
Like terms are terms with the same variable (or no variable, i.e., constants). Combine them by adding or subtracting their coefficients to simplify algebraic expressions, such as combining constant terms to find the y-intercept in slope-intercept form.
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