Question

select all of the equations that represent linear relationships.
□ 5 + 2y = 13
□ $y = \frac{1}{2}x^2 + 7$
□ $y - 5 = 2(x - 1)$
□ $\frac{y}{2} = x + 7$
□ $x = -4$
done

Explanation:

Step1: Recall linear equation form

A linear equation is of the form \(y = mx + b\) (slope - intercept), \(Ax+By = C\) (standard), or \(y - y_1=m(x - x_1)\) (point - slope), and the highest power of the variable is 1.

Step2: Analyze \(5 + 2y=13\)

We can rewrite it in standard form. Subtract 5 from both sides: \(2y=13 - 5=8\), then \(y = 4\). This is a horizontal line (a linear equation with slope \(m = 0\)). So it represents a linear relationship.

Step3: Analyze \(y=\frac{1}{2}x^{2}+7\)

The highest power of \(x\) is 2, so this is a quadratic equation (a parabola), not a linear relationship.

Step4: Analyze \(y - 5=2(x - 1)\)

This is in point - slope form (\(y - y_1=m(x - x_1)\) where \(y_1 = 5\), \(x_1=1\), \(m = 2\)). When we expand it: \(y-5 = 2x-2\), then \(y=2x + 3\), which is in slope - intercept form. So it represents a linear relationship.

Step5: Analyze \(\frac{y}{2}=x + 7\)

Multiply both sides by 2: \(y=2x + 14\), which is in slope - intercept form (\(y=mx + b\) with \(m = 2\), \(b = 14\)). So it represents a linear relationship.

Step6: Analyze \(x=-4\)

This is a vertical line (a linear equation with an undefined slope). It represents a linear relationship.

Answer:

• \(5 + 2y = 13\)
• \(y - 5 = 2(x - 1)\)
• \(\frac{y}{2}=x + 7\)
• \(x=-4\)