Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (___, 3).
2x + y = 1
x² + y² = 10

검증된 단계별 안내

Start by substituting the given y-value into the first equation. Since one solution has y = 3, plug y = 3 into the linear equation \(2x + y = 1\) to find the corresponding x-value.

Rewrite the first equation with y = 3: \(2x + 3 = 1\). Then, solve for \(x\) by isolating it on one side: \(2x = 1 - 3\).

Simplify the right side to get \(2x = -2\), and then divide both sides by 2 to find \(x = -1\).

Now, verify this solution by substituting \(x = -1\) and \(y = 3\) into the second equation \(x^2 + y^2 = 10\) to ensure it satisfies the equation.

To find the other solution, use the system of equations: express \(y\) from the first equation as \(y = 1 - 2x\), then substitute this expression into the second equation \(x^2 + y^2 = 10\) and solve the resulting quadratic equation for \(x\). Finally, find the corresponding \(y\) values.

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Solving Systems of Equations

A system of equations consists of two or more equations with the same variables. Solving the system means finding all variable values that satisfy every equation simultaneously. Methods include substitution, elimination, and graphing, which help find points where the equations intersect.

Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one equation is linear and the other nonlinear.

Nonlinear Equations and Circles

Nonlinear equations, such as circles defined by x² + y² = r², represent curves rather than lines. Understanding the geometric meaning helps interpret solutions as intersection points between a line and a curve. These intersections correspond to the system's solutions.

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (___, 3). 2x + y = 1 x2 + y2 = 10