Determine the system of equations illustrated in each graph. Write equations in standard form.

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Identify the two lines and their key points from the graph. The first line passes through points (0, 7) and (-4, 0). The second line passes through points (0, 2) and (6, 0).

Find the slope of the first line using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using points (0, 7) and (-4, 0), calculate \(m_1 = \frac{0 - 7}{-4 - 0} = \frac{-7}{-4} = \frac{7}{4}\).

Use the point-slope form of a line equation \(y - y_1 = m(x - x_1)\) with point (0, 7) and slope \(\frac{7}{4}\) to write the equation of the first line: \(y - 7 = \frac{7}{4}(x - 0)\).

Convert the first line's equation to standard form \(Ax + By = C\) by multiplying both sides to clear fractions and rearranging terms.

Repeat the process for the second line: find slope \(m_2 = \frac{0 - 2}{6 - 0} = \frac{-2}{6} = -\frac{1}{3}\), use point-slope form with point (0, 2), then convert to standard form.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Finding the Equation of a Line from Two Points

To find the equation of a line, use two points on the line to calculate the slope (m) as the change in y over the change in x. Then apply the point-slope form y - y1 = m(x - x1) to write the equation. This method is essential for determining the line's equation from graph points.

Converting to Standard Form of a Linear Equation

Standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A ≥ 0. After finding the slope-intercept form, rearrange terms to get all variables on one side and the constant on the other. This form is often required for system of equations problems.

Interpreting Graphs to Identify Key Points

Graphs provide visual data such as intercepts and points where lines cross. Identifying these points accurately is crucial for writing equations. For example, x- and y-intercepts give direct points to use in slope calculations and equation formulation.