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Section2.1Equation of a line: standard form

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Subsection2.1.1Activities and Problems

Determine, by hand, the \(x\)-intercept and \(y\)-intercept of the line with equation \(2x-3y=6\text{.}\) Remember to state the intercepts as ordered pairs. Check you answer by using the Desmos sliders to set \(a\) to \(2\text{,}\) \(b\) to \(-3\text{,}\) and \(c\) to \(6\text{.}\)

Solution

To determine the \(x\)-intercept (where the line crosses the \(x\)-axis), we replace \(y\) with \(0\) and solve for \(x\text{.}\) Solving \(2x-3(0)=6\) results in \(3\text{,}\) so the \(x\)-intercept is \((3,0)\text{.}\)

To determine the \(y\)-intercept (where the line crosses the \(y\)-axis), we replace \(x\) with \(0\) and solve for \(y\text{.}\) Solving \(2(0)-y=6\) results in \(-2\text{,}\) so the \(y\)-intercept is \((0,-2)\text{.}\)

Determine, by hand, the \(x\)-intercept and \(y\)-intercept of the line with equation \(4x+y=4\text{.}\) Remember to state the intercepts as ordered pairs. Check you answer by using the desmos sliders to set \(a\) to \(4\text{,}\) \(b\) to \(1\text{,}\) and \(c\) to \(4\text{.}\)

Solution

The \(x\)-intercept is \((1,0)\) and the \(y\)-intercept is \((0,4)\)

Determine, by hand, the \(x\)-intercept and \(y\)-intercept of the line with equation \(-x-4y=10\text{.}\) Remember to state the intercepts as ordered pairs. Check you answer by using the desmos sliders to set \(a\) to \(-1\text{,}\) \(b\) to \(-4\text{,}\) and \(c\) to \(10\text{.}\)

Solution

The \(x\)-intercept is \((-10,0)\) and the \(y\)-intercept is \((0,-\frac{5}{2})\)

Set the value of \(a\) to \(0\) and the value of \(b\) to \(1\text{.}\) Note that this results in equations of form \(y=c\text{.}\) Slide the value of \(c\) back and forth. What do all lines whose equations can be expressed in the form \(y=c\) have in common?

Solution

Every line whose equation can be expressed in the form \(y=c\) is horizontal.

Set the value of \(a\) to \(1\) and the value of \(b\) to \(0\text{.}\) Note that this results in equations of form \(x=c\text{.}\) Slide the value of \(c\) back and forth. What do all lines whose equations can be expressed in the form \(x=c\) have in common?

Solution

Every line whose equation can be expressed in the form \(x=c\) is vertical.