Question Rewrite the equation $y-8=-\frac{1}{2}(x-3)$ to standard form.

Studdy Solution

STEP 1

Assumptions
1. The given equation is in point-slope form: $y - y_1 = m(x - x_1)$
2. The standard form of a linear equation is $Ax + By = C$ , where $A$ , $B$ , and $C$ are integers, and $A$ is non-negative.

STEP 2

We start by expanding the right-hand side of the equation to get rid of the parentheses.
$y - 8 = -\frac{1}{2}(x - 3)$

STEP 3

Distribute the $-\frac{1}{2}$ across the terms inside the parentheses.
$y - 8 = -\frac{1}{2}x + \frac{1}{2} \cdot 3$

STEP 4

Simplify the right-hand side of the equation.
$y - 8 = -\frac{1}{2}x + \frac{3}{2}$

STEP 5

To convert to standard form, we want to have $x$ and $y$ on the same side of the equation. So, we will add $\frac{1}{2}x$ to both sides of the equation.
$\frac{1}{2}x + y - 8 = \frac{3}{2}$

STEP 6

Next, we want to eliminate the fractions to have integer coefficients. To do this, we can multiply every term by the denominator of the fractions, which is 2.
$2 \cdot \left(\frac{1}{2}x\right) + 2 \cdot y - 2 \cdot 8 = 2 \cdot \frac{3}{2}$

STEP 7

Simplify the equation by carrying out the multiplications.
$x + 2y - 16 = 3$

STEP 8

Now, we need to move the constant term to the right side of the equation to match the standard form. We do this by adding 16 to both sides of the equation.
$x + 2y = 3 + 16$

STEP 9

Simplify the right-hand side of the equation.
$x + 2y = 19$

STEP 10

The equation is now in standard form, but we need to ensure that the coefficient of $x$ is non-negative. Since it is already non-negative, we do not need to make any further changes.
The equation in standard form is:
$x + 2y = 19$

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